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%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{Extracting and Unifying Semi-Elasticities and Effect Sizes from Studies with Binary Dependent Variables}
\title{ Extracting and Unifying Semi-Elasticities\\
and Effect Sizes from Studies\\
with Binary Dependent Variables }
\date{\vspace{-5ex}}
\author{
{Geraldine Henningsen}%
\footnote{corresponding author.
E-mail: \url{geraldine.henningsen@googlemail.com}.}
\and
{Arne Henningsen}%
\footnote{
University of Copenhagen,
Department of Food and Resource Economics,
Rolighedsvej~23, 1958~Frederiksberg~C, Denmark.
Phone: +45-3533-2274, e-mail: \url{arne@ifro.ku.dk}.}
\thanks{Geraldine Henningsen was supported by the `SAVE-E' project funded by the
`Innovation Fund Denmark' (grant number:~4106-00009B).
The authors declare no conflicts of interest.
Senior authorship is shared.}
}
\begin{document}
%\SweaveOpts{concordance=TRUE}
\maketitle
\begin{abstract}
Combining the results from regression analyses in a meta-analysis
often proves difficult
because differences in the statistical methods and in the units of measurement
or the encoding of the variables invalidate a direct comparison of
the regression coefficients.
In this article, we suggest simple and straightforward methods to extract
unified measures for quantifying effects on binary dependent variables
that are comparable across different studies
even if the studies use different statistical methods,
different units of measurement, or different codings of the variables.
The suggested effect measures can be applied to continuous,
interval-coded, and categorical covariates.
We, furthermore, suggest methods to obtain valid approximations of the standard errors
of the unified effect measures that can be used, e.g.,
as weighting factors in a subsequent meta-analysis.
We have implemented all suggested methods
in the \R{} package \pkg{urbin}
that we use to demonstrate the application of our methodology.
\bigskip
Keywords: binary dependent variables, meta-analysis, logit, probit
\end{abstract}
%===================================== Introduction =======================================
\section{Introduction}
\label{sec:intro}
Combining the results from different regression analyses in a meta-analysis
often proves difficult because differences in the applied estimation methods and
differences in the units of measurement or the encoding
of the variables of interest invalidate a direct comparison of the regression coefficients.
For simple linear regression models, e.g., ordinary least squares regression models,
these problems can to some extent be overcome
by calculating an `elasticity' for each continuous covariate of interest at the sample mean,
and a relative effect size of each categorical covariate of interest.
These measures indicate by how many percent the dependent variable changes
if a continuous covariate increases by one percent
or if a categorical covariate changes from a reference category
to the category of interest.
However, for many non-linear regression models,
e.g., generalised linear models,
this approach is no longer straightforward and often requires more statistical information
than is usually provided in articles,
in particular if the user wishes to obtain valid standard errors for those measures.
This article introduces simple and straightforward methods to extract comparable effect measures
and their corresponding standard errors from studies with binary and categorical dependent variables.%
\footnote{The estimation methods covered here are the linear probability model,
logistic regression, probit regression,
ordinal probit regression, multivariate probit regression,
and multinomial logistic regression.
We refrain for the time being from estimation methods for discrete choice experiments,
e.g., conditional logistic regression, or mixed logistic regression.
}
We demonstrate how to derive semi-elasticities for continuous covariates and
effect size measures for categorical or ordered covariates from the statistical information
usually provided in articles.
Furthermore, we demonstrate how to transform and unify differently encoded variables,
by showing how to calculate a semi-elasticity of an interval-coded covariate,
how to calculate the effect size by turning a continuous
covariate into an interval-coded covariate,
and how to change the reference category or
the grouping of a categorical covariate in order to make effects comparable across
different studies.
Finally, we introduce a simple and novel way to calculate valid approximations
of the standard errors for the derived semi-elasticities and effect size measures
in cases where the full variance-covariance matrix of the regression model is unavailable%
---which we deem to be the standard for most publications.
We demonstrate the application of our methodology by means of a data set
on women's labour force participation \cite{Mroz1987}
and the \R{} \cite{RCT2018} package \pkg{urbin} \cite{r-urbin},
in which we have implemented all methods that we suggest in this article.
The article is structured as follows: section two gives a brief introduction to the
data set;
section three briefly presents the regression methods
that we cover in this article;
sections four to seven discuss the various approaches for calculating semi-elasticities,
effect sizes, and corresponding standard errors;
section eight demonstrates how these approaches can be applied
to non-binary categorical dependent variables;
finally, section nine concludes.
%======================================= Data =========================================
\section{Data for empirical example}
\label{sec:data}
We use an empirical example
based on a data set on women's labour force participation \cite{Mroz1987}
to demonstrate the application of our methodology using \R{} package
\pkg{urbin} \cite{r-urbin}
and to test the validity of the approximated standard errors in cases where the
full variance-covariance matrix is unavailable to the user.
The data set is available through the \R{} package \pkg{sampleSelection} \cite{Toomet2008}
under the name~\code{Mroz87}.
<<dataLoad,results='hide', echo=FALSE>>=
data( "Mroz87", package = "sampleSelection" )
@
The data set contains 753~observations on married women and their respective
labour force participation in the year 1975,
as well as various socio-economic background variables.
In total the data set includes 22~variables.
Table~\ref{tab:descStats} provides the summary statistics
of the variables in this data set.%
\footnote{%
A more detailed description of the variables in this data set
is available, e.g., in the documentation of the \pkg{sampleSelection}
package.
The \R{} code that loads the data set and prepares it
for the examples in Sections~\ref{sec:semela} to~\ref{sec:non-binaryDepVar}
is available in Appendix Section~\ref{sec:RCodeData}.
}
%%% moved this code here so that 'kids', 'lfp3', and the variables with age
%%% categories are included in the table with descriptive statistics
<<dataLfp3, echo=FALSE, results='hide'>>=
Mroz87$lfp3 <- factor( ifelse( Mroz87$hours == 0, "no",
ifelse( Mroz87$hours <= 1300, "part", "full" ) ),
ordered = TRUE, levels = c( "no", "part", "full" ) )
@
<<dataKids, echo=FALSE, results='hide' >>=
Mroz87$kids <- Mroz87$kids5 + Mroz87$kids618
@
<<dataAgeInt, echo=FALSE, results='hide' >>=
Mroz87$age30.37 <- Mroz87$age >= 30 & Mroz87$age <= 37
Mroz87$age38.44 <- Mroz87$age >= 38 & Mroz87$age <= 44
Mroz87$age45.52 <- Mroz87$age >= 45 & Mroz87$age <= 52
Mroz87$age53.60 <- Mroz87$age >= 53 & Mroz87$age <= 60
all.equal(
Mroz87$age30.37 + Mroz87$age38.44 + Mroz87$age45.52 + Mroz87$age53.60,
rep( 1, nrow( Mroz87 ) ) )
@
<<tabDescStat,results='asis', echo=FALSE, message = FALSE >>=
library( "stargazer" )
stargazer( Mroz87, title = "Descriptive Statistics",
label = "tab:descStats",
float.env = "sidewaystable", header = FALSE, digits = 2, align = TRUE,
omit.summary.stat = c( "p25", "p75" ) )
@
In our empirical example,
we use the women's labour force participation (\code{lfp}) as dependent (outcome) variable.
Variable \code{lfp} is a dummy variable, where a `1' represents
labour force participation and a `0' represents no labour force participation.
We regress this variable
on a dummy variable for the presence of children in the household (\code{kids}),
the woman's age in years (\code{age}), and her years of education (\code{educ}):
\begin{equation}
\Pr( \mathtt{lfp} = 1 | \mathtt{kids}, \mathtt{age}, \mathtt{educ} )
= f( \mathtt{kids}, \mathtt{age}, \mathtt{educ} )
\label{eq:esti}
\end{equation}
Variable \code{age} is our primary variable of interest
and we use it either as a continuous covariate
or as an interval-coded covariate with four intervals:
30--37, 38--44, 45--62, and 63--70~years.
To demonstrate how to apply our methods
to regression models where the dependent variable has more than two categories,
e.g., ordered probit models and multinomial logistic regressions,
and to test approximations of standard errors from estimates derived from
these regressions models,
we create an additional variable for labour force participation
that has three categories: `no labour force participation' (0~working hours),
`part-time labour force participation' (1--1,300~working hours),
and `full-time labour force participation' (>1,300~working hours).
We estimate equation~(\ref{eq:esti}) with the estimation methods
discussed in this article.
The regression results provide the variance-covariance matrices
of the estimated coefficients
so that we can apply the Delta method \cite{Greene2011}
to calculate approximate standard errors of the calculated
effect size measures.
We use these standard errors as benchmarks
to assess the quality of various approximations for cases
where all off-diagonal elements of the variance-covariance matrix are unknown,
which is the case for most studies published in the literature,
as usually only the standard errors (or t-values) of the estimates are reported.
%================================= Estimation Methods =====================================
\section{Estimation methods}
\label{sec:method}
Most estimation methods that can handle binary or categorical
dependent variables can be categorised
into two groups: methods where the link function follows a normal distribution,
so-called probit regressions, and methods where the link function follows a
logistic distribution, logistic regressions.%
\footnote{There exists a multitude of estimation techniques for models with
categorical dependent variables, quasi-categorical dependent variables,
like count data, or outcome variables that can be transferred into a binary
or categorical variables, like truncated variables.
We consider these regression models to be outside the scope of this article.}
Another approach that has regained popularity in recent years
because it is based on fewer assumptions than other approaches
is the linear probability model,
which uses a simple linear link function,
and, thus, can be estimated by ordinary least squares (OLS)
and other estimators for linear regression models.
The linear probability model assumes that a Bernoulli trial can be explained
by a linear combination of covariates:
\begin{equation}
\Pr( Y = 1 | X = x ) = \beta_0 + \sum_{j=1}^K \beta_j x_j,
\label{eq:lpm}
\end{equation}
where $Y \in \{ 0, 1 \}$ is a binary dependent variable,
$x = ( x_1, \ldots , x_K )^{\top}$ is a vector
of $K$ covariates, and
$\beta = ( \beta_0 , \ldots , \beta_K )^{\top}$ is a vector of $K + 1$
unknown coefficients.
A probit regression model \cite{Bliss1934, Fisher1935}
models the same relationship assuming a probit link function
which follows a standard normal distribution:
\begin{equation}
\Pr( Y = 1 | X = x ) = \Phi\left( \beta_0 + \sum_{j=1}^K \beta_j x_j \right) ,
\label{eq:probit}
\end{equation}
where $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution.
The logistic regression \cite{Cox1958} uses a logit link function:
\begin{equation}
\Pr( Y = 1 | X = x ) = \frac{ \exp \left( \beta_0 + \sum_{k=1}^K \beta_k x_k \right)}
{ 1 + \exp \left( \beta_0 + \sum_{k=1}^K \beta_k x_k \right)},
\label{eq:logit}
\end{equation}
A bivariate or multivariate probit model generalises the probit regression model~(\ref{eq:probit})
to simultaneously estimate two or more probit equations for different binary dependent variables
$Y_1 , \ldots , Y_N$,
where a potential correlation between the error terms of the different probit equations
is explicitly modelled.
As meta-analyses usually focus on one specific dependent variable,
the coefficients of the regression equations for the other dependent variables and
the correlation structure of the error terms in the multivariate probit regression model
can be ignored.
Hence, the estimation result for the one probit equation of interest
can be treated like an estimation result from a univariate probit regression,
so that equation~(\ref{eq:probit}) can be used to calculate
the \textit{unconditional} probabilities
$P( Y_n = 1 | x_1 , \ldots , x_K )$
and the marginal effects on the \textit{unconditional} probabilities
in bivariate and multivariate probit regression models \cite{Sodjinou2012}.%
\footnote{%
We do not take into account the conditional probabilities
$P( Y_n = 1 | x_1, \ldots, x_K,\allowbreak Y_1, \ldots, Y_{n-1},\allowbreak Y_{n+1}, \ldots, Y_P)$
and the marginal effects on the conditional probabilities,
because we focus on one dependent variable
and disregard interrelations between different dependent variables.
}
If a study reports the marginal effects based on the regression results
of a probit model, a logistic regression, or a multivariate probit model,
one can assume a first-order Taylor series approximation of these models
around the point, at which the marginal effects were calculated.
Under this linear approximation, the marginal effects can be treated
as if they were coefficients of a linear probability model~(\ref{eq:lpm}).
Many extensions of probit and logistic regression models have been developed to accommodate for
more complicated data structures.
For instance, regression methods for dependent variables with more than two categories
estimate the probability
that the dependent variable is equal to a certain category.
Studies with this set-up can still be compared to studies with a binary dependent variable
if the categories can be grouped into two groups
that correspond to the two outcomes of the binary dependent variable
in the other studies.
A modification of the probit regression that handles ordered categorical variables,
i.e., categorical variables where the ordering of the categories is meaningful
(think of first place, second and third place),
is the ordered probit regression \cite{Aitchison1957},
where the dependent variable can have $P$~distinct and strictly ordered values
($Y \in \{ 1, \ldots , P \}$),
can be specified as:
\begin{align}
\Pr( Y = p | X = x )
= \; & \Phi\left( \mu_p - \sum_{j = 1}^K \beta_j x_j \right)
- \Phi\left( \mu_{p - 1} - \sum_{j = 1}^K \beta_j x_j \right)
\label{eq:oprobit} \\
& \forall \; p = 1, \ldots, P, \nonumber
\end{align}
where $\mu_0 < \mu_1 < \ldots < \mu_P$ are the break points,
of which $\mu_0 = - \infty$, $\mu_P = \infty$,
and $\mu_1 , \ldots , \mu_{P-1}$ are unknown and, thus, need to be estimated.
%
To make estimates from an ordered probit model comparable to
estimates from models with a binary dependent variable,
we create a new binary dependent variable~$Y^*$
by dividing the $P$~distinct values of the dependent variable~$Y$
into two categories:
\begin{equation}
Y^* =
\begin{cases}
0 & \text{ if } Y \in \{ 1, \ldots , p^* - 1 \} \\
1 & \text{ if } Y \in \{ p^* , \ldots , P \}
\end{cases}
\end{equation}
with $p^* \in \{ 2, \ldots, P \}$.
This reduces the ordered probit model to a binary probit model:%
\footnote{%
A proof is given in Appendix Section~\ref{sec:derivBinProbitOrderedProbit}.
}
\begin{align}
\Pr( Y^* = 1 | X = x )
& = \Pr( Y \in \{ p^*, \ldots , P \} | X = x ) \label{eq:prob1} \\
& = \Phi\left( - \mu_{p^* - 1} + \sum_{j = 1}^K \beta_j x_j \right) \label{eq:probn},
\end{align}
where the intercept of the binary probit model~(\ref{eq:probit})
is equal to the negative value of the break point
that separates $Y^* = 0$ from $Y^* = 1$,
i.e., $\beta_0 = -\mu_{p^* - 1}$.%
\footnote{%
If the ordered probit model~(\ref{eq:oprobit})
is estimated with intercept, say, $\beta_0^*$ and
(for identification) by normalising the first (internal) break point to zero,
i.e., $\mu_1 = 0$,
the ordered probit model can be simplified to a binary probit model
with intercept $\beta_0 = \beta_0^* - \mu_{p^* - 1}$.
}
The multinomial logistic regression \cite{Luce1959} handles estimation models with
multinomial dependent variables, i.e., categorical variables where the ordering
of the categories has no meaning (think of red cars, green, cars, and blue cars):
\begin{align}
\Pr( Y = p | X = x ) & \equiv \pi_p\\
& = \frac{ \exp \left( \beta_{0,p} + \sum_{k=1}^K \beta_{k,p} x_k \right) }
{ 1 + \sum_{o \in \{1, \ldots, P \} \setminus p^* }
\exp \left( \beta_{0,o} + \sum_{k=1}^K \beta_{k,o} x_k \right) } \\[1ex]
& = \frac{ \exp \left( \beta_{0,p} + \sum_{k=1}^K \beta_{k,p} x_k \right) }
{ \sum_{o=1}^P
\exp \left( \beta_{0,o} + \sum_{k=1}^K \beta_{k,o} x_k \right) } \\[1ex]
& \hspace{2cm} \forall \, p = 1, \ldots, P, \nonumber
\label{eq:MNLogit}
\end{align}
where $Y \in \{1, \ldots, P \}$ is a categorical dependent variable
with reference category~$p^*$,
$\beta_{\cdot,p} = ( \beta_{0, p}, \ldots, \beta_{K, p} )^{\top}$;
$p \in \{1, \ldots, P \} \setminus p^*$
are $P-1$ vectors of $K + 1$ unknown coefficients each,
and $\beta_{j,p^*} \equiv 0 \; \forall \; j = 0, \ldots, K$
are the $K+1$ coefficients of the reference category~$p^*$,
which are all normalized to zero.
%=============================== Continuous Covariates ===================================
\section{Semi-elasticities of continuous covariates}
\label{sec:semela}
\subsection{Semi-elasticities}
Differences in the units of measurement of the variables of interest often
render it impossible to directly compare coefficient estimates from
different studies.
In many cases,
this problem can be circumvented by calculating the elasticity of the effect,
$\epsilon_k = \partial \ln( Y ) / \partial \ln( x_k )
= ( \partial Y / \partial x_k ) \cdot ( x_k / Y )$,
e.g., calculated at the sample mean $x_k = \bar{x}_k$
and $Y = \bar{Y}$ (e.g., see \cite{Greene2010}).
The elasticity describes the percentage change in the dependent variable~$Y$
given a one percent increase in the continuous covariate~$x_k$.
It is as such unit-free,
which allows the user to compare the effect of a particular covariate across different studies.
In the case of studies with binary dependent variables,
the regression model describes the probability
that the dependent variable has a value of one.
As the probability is already coded between zero and one and unit-free,
we suggest to calculate a semi-elasticity of each continuous
covariate of interest:
\begin{equation}
\epsilon_k \equiv \frac{\partial \Pr( Y = 1 | X = x )}{\partial x_k } \cdot x_k ,
\label{eq:semiEl}
\end{equation}
which can be interpreted as the percentage point change in the probability of $Y$
being equal to one given a one percent increase in~$x_k$.
Table~\ref{tab:semiE} presents the equations for calculating the semi-elasticities
of continuous covariates for all six estimation methods covered in this article.
If the covariate of interest enters the estimation
equation both as linear term and as quadratic term,
the equations for calculating the semi-elasticities
must be extended accordingly.
These extensions for quadratic terms are included in the equations
in Table~\ref{tab:semiE}.
\newcommand{\noteMLogitP}{%
For multinomial logistic regression models,
$\mathcal{P}$ indicates the set of categories of the dependent variable
that correspond to a binary outcome of one,
while all categories that are not in $\mathcal{P}$
correspond to a binary outcome of zero.
}
\begin{sidewaystable}[hp]
\caption{Semi-elasticities of continuous covariates for 6~different regression models}
\label{tab:semiE}
\begin{tabular}{ p{5cm} p{15cm} }
\\[-1.8ex]\hline
\hline \\[-1.8ex]
Regression model & Semi-elasticity \\
\midrule \\[-1.8ex]
Linear probability model &
\begin{equation*}
\epsilon_k = \left( \beta_k + 2 \, \beta_{k+1} x_k \right) x_k
\end{equation*} \\
Binary, bivariate / multivariate, and ordered probit regression &
\begin{equation*}
\epsilon_k =
\phi \left( \beta_0 + \sum_{j = 1}^K \beta_j x_j \right)
\left( \beta_k + 2 \, \beta_{k+1} x_k \right) x_k
\end{equation*}\\
Logistic regression &
\begin{equation*}
\epsilon_k =
\dfrac{ \exp{( \beta_0 + \sum_{j=1}^K \beta_j x_j )}}
{( 1 + \exp{( \beta_0 + \sum_{j=1}^K \beta_j x_j )})^2 }
\left( \beta_k + 2 \, \beta_{k+1} x_k \right) x_k
\end{equation*}\\
Multinomial logistic regression &
\begin{equation*}
\begin{aligned}
\epsilon_{k,\mathcal{P}}
= \; & \sum_{p \in \mathcal{P}} \epsilon_{k,p}^*\\
\text{with } \epsilon_{k,p}^*
= \; & \pi_p \left(
\beta_{k,p} + 2 \, \beta_{k+1,p} x_k
- \sum_{o=1}^P
\left( \beta_{k,o} + 2 \, \beta_{k+1,o} x_k \right)
\pi_o \right) x_k \\
& \forall \; p = 1, \ldots, P
\end{aligned}
\end{equation*} \\[1em]
\bottomrule
\end{tabular}\\[1ex]
Note:
$\phi(\cdot)$ denotes the probability density function
of the standard normal distribution.
If the regression analysis includes a quadratic term
of the covariate of interest,
we assume that the model is specified with $x_{k+1} \equiv x_k^2$
so that $\beta_{k+1}$ and $\beta_{k+1,j}; j = 1, \ldots, P$ indicate
the coefficient(s) of the quadratic term of the covariate of interest.
If the regression analysis does not include a quadratic term
of the covariate of interest,
the equations in this table can be simplified
by setting all $\beta_{k+1}$ and all $\beta_{k+1,j}; j = 1, \ldots, P$
to zero.
\noteMLogitP
\end{sidewaystable}
As described in Section~\ref{sec:method},
certain estimates from bivariate, multivariate, and ordered probit regressions
can be extracted
so that parts of these models correspond to binary probit models
and the equation for calculating semi-elasticities of binary probit models
can be also applied to bivariate, multivariate, and ordered probit models.
We have implemented the calculation of semi-elasticities
for all six estimation methods covered in this article
in the \R{} package \pkg{urbin}.
In order to demonstrate how to use this package,
we calculate the semi-elasticity of the variable \code{age} with regard to
a married woman's probability to participate in the labour force
based on a probit regression~(\ref{eq:probit}) of equation~(\ref{eq:esti}).%
\footnote{%
The \R{} code for estimating these models
is available in Appendix Section~\ref{sec:RCodeEstProbit}.
}
<<estProbit, echo=FALSE, results='hide' >>=
estProbit <- glm( lfp ~ kids + age + educ,
family = binomial(link = "probit"), data = Mroz87 )
xMean <- c( 1, colMeans( Mroz87[ , c( "kids", "age", "educ" ) ] ) )
estProbitQ <- glm( lfp ~ kids + age + I(age^2) + educ,
family = binomial(link = "probit"), data = Mroz87 )
xMeanQ <- c( xMean[ 1:3], xMean[3]^2, xMean[4] )
@
<<tabEstProbit, echo=FALSE, results='asis' >>=
stargazer( estProbit, estProbitQ,
title = "Probit regression results with age as linear and quadratic covariate",
label = "tab:probitSum", header = FALSE,
digits = 2, intercept.bottom = FALSE )
@
The results of the probit regression are presented in Table~\ref{tab:probitSum},
which in combination with Table~\ref{tab:descStats} represents the standard
information that a user usually can obtain from most publications.
In the following command,
function \code{urbinEla} calculates the semi-elasticity of variable \code{age}:
<< echo=FALSE, results='hide' >>=
library( "urbin" )
@
<<>>=
urbinEla( coef(estProbit), xMean, xPos = 3, model = "probit" )
@
This is done based on the vector of coefficients (including intercept)
of the probit regression, \code{coef(estProbit)},
and the vector of sample means for all covariates
(including a one for the intercept), \code{xMean}.
Argument \code{xPos} indicates the position of the covariate(s) of interest,
in our example \code{age},
in the vectors \code{coef(estProbit)} and \code{xMean}.
Argument \code{model} is set to \code{"probit"}, because the coefficients are
obtained from a probit regression and, thus, the semi-elasticity of
variable \code{age} has to be calculated based on equation~(\ref{eq:probit}).
The calculated semi-elasticity indicates
that the probability that a woman is in the labour force decreases,
\textit{ceteris paribus}, by
\Sexpr{-round(urbinEla( coef(estProbit), xMean, xPos = 3, model = "probit" )$semEla, 2 )}%
~percentage points if her age increases by one percent.
If variable \code{age} also enters the regression equation in quadratic form,
we can simply use argument \code{xPos} to point out the positions
of both \code{age} and \code{age}$^2$
in vectors \code{coef(estProbitQ)} and \code{xMeanQ}:
<<>>=
urbinEla( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
model = "probit" )
@
If argument \code{xPos} has two elements,
function \code{urbinEla} automatically uses the extended
formula to accommodate the quadratic term in the calculation of the
semi-elasticity.
The semi-elasticity
based on the probit model with both a linear and a quadratic term of \code{age}
indicates
that the probability that a woman is in the labour force decreases,
\textit{ceteris paribus}, by
\Sexpr{-round(urbinEla( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ), model = "probit" )$semEla, 2 )}%
~percentage points if her age increases by one percent.
\subsection{Approximation of standard errors}
\label{sec:semElaSE}
An approximate standard error of the semi-elasticity
defined in equation~(\ref{eq:semiEl})
can be obtained by using the Delta method \cite{Greene2011}:
\begin{equation}
\se( \epsilon_k )
= \sqrt{
\frac{ \partial \epsilon_k }{ \partial \bm{\beta} }
\var( \bm{\beta} )
\frac{ \partial \epsilon_k }{ \partial \bm{\beta}^{\top} } },
\label{eq:DeltaMethod}
\end{equation}
where $\se( \epsilon_k )$ indicates
the (approximate) standard error of the semi-elasticity $\epsilon_k$,
$\partial \epsilon_k / \partial \bm{\beta}$
indicates the gradient vector of the semi-elasticity $\epsilon_k$
with respect to the coefficients $\beta_0, \ldots, \beta_K$,
and $\var( \bm{\beta} )$ indicates
the variance-covariance matrix of the estimated coefficients.
The gradient vectors for the semi-elasticities,
$\partial \epsilon_k / \partial \bm{\beta}$,
of the various regression models
are presented in Appendix Section~\ref{sec:gradSemEla}.
The following commands calculate the same semi-elasticities as above,
but this time include their respective standard errors
based on the full variance-covariances matrices of the estimates:
<<>>=
urbinEla( coef(estProbit), xMean, xPos = 3, model = "probit",
allCoefVcov = vcov(estProbit) )
urbinEla( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
model = "probit", allCoefVcov = vcov(estProbitQ) )
@
As scientific publications usually do not report covariances
between estimated coefficients,
but only (at best) standard errors
(or t-values, which can be used to calculate the standard errors),
the covariances between the estimates of the coefficient
are usually unknown.
A simple solution would be to replace the unknown covariances by zeros.
However, in many empirical examples and a few Monte-Carlo trials,
we noticed that ignoring the covariances between the coefficients
often gives very imprecise, mostly upward-biased, standard errors
of the semi-elasticities,
particularly if the models include a quadratic term of the covariate of interest:
<<>>=
urbinEla( coef(estProbit), xMean, xPos = 3, model = "probit",
allCoefVcov = sqrt(diag(vcov(estProbit))),
seSimplify = FALSE )
urbinEla( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
model = "probit", allCoefVcov = sqrt(diag(vcov(estProbitQ))),
seSimplify = FALSE )
@
In these empirical examples and Monte Carlo trials,
we found that when covariances are assumed to be zero,
simplifying the calculations of the gradients
$\partial \epsilon_k / \partial \bm{\beta}$
by assuming that the `weighting factors' in the equations
for calculating the semi-elasticities%
\footnote{I.e., $\phi(\cdot)$ for different types of probit models,
$\exp(\cdot)/(1+\exp(\cdot))^2$ for logistic regression models, and
$\pi_p$ and $\pi_o$ for multinomial logistic regression models,
see Table~\ref{tab:semiE}.}
do not depend on the coefficients (although they actually do),
gives much better approximations of the standard error
than using the correctly calculated gradients.
These simplified gradient vectors are presented
in Appendix Section~\ref{sec:gradSemElaSimple}.
As several elements of these simplified gradient vectors are zero,
the calculation of the semi-elasticities with the Delta method
ignores many of the unknown covariances
so that a lack of covariances causes a smaller problem
when using the simplified gradients than when using the full gradients.
The huge overestimation of the standard errors of the semi-elasticities
in the presence of a quadratic term of the covariate of interest
originates from the multicollinearity between the quadratic term,
the linear term, and the intercept.
In an OLS regression, e.g., a linear probability model,
with an intercept and a linear and quadratic term of a covariate,
the variance-covariance matrix of the estimates
would be equal to $\sigma^2 ( X^{\top} X )^{-1}$,
where $\sigma^2$ is the variance of the error term
and $X$ is an $N \times 3$ matrix with $N$ the number of observations
and its three columns being the intercept and the linear and quadratic term
of the covariate, respectively.
If we have the values of the covariate,
we can calculate the elements~$w_{ij}; i,j \in \{1,2,3\}$
of the $3 \times 3$ matrix $W \equiv ( X^{\top} X )^{-1}$.
If we additionally have the standard errors of the coefficients
of the linear and quadratic terms of the covariate,
i.e., $\se(\beta_1)$ and $\se(\beta_2)$, respectively,
we can calculate the variance of the error term
as $\sigma^2 = \se(\beta_1)^2 / w_{22}$
or as $\sigma^2 = \se(\beta_2)^2 / w_{33}$
and then the covariance between the two coefficients of the covariate
as $\cov(\beta_1, \beta_2) = \sigma^2 w_{23} = \sigma^2 w_{32}$.
As one usually does not have the original data that were used in published studies,
but rather the mean value and corresponding standard deviation of the covariate of interest,
one can simulate the values of the covariate, e.g.,
with a pseudo-random number generator sampling from a normal distribution
using the actual mean and standard deviation of the covariate.
In cases where the covariate is simulated,
the actual model includes further covariates, or
the actual model is not an OLS model (but, e.g., a probit or logit regression),
the two above-described equations for calculating the variance of the error term
give two different values.
In these cases, one can calculate the approximate error variance
as a geometric mean:
$\sigma^2 \approx \sqrt{ \left( \se(\beta_1)^2 / w_{22} \right)
\left( \se(\beta_2)^2 / w_{33} \right) }$,
which is a more conservative measure than the arithmetic mean.
If a user of the \code{urbinEla} function provides standard errors
of the coefficients (rather than the full covariance matrix),
it uses the simplified gradients to calculate the standard errors
unless the user sets argument \code{seSimplify} to \code{FALSE}.
Moreover, if the model includes a quadratic term
of the covariate of interest
and the user provides the mean value and the standard deviation
of the covariate of interest through argument \code{xMeanSd},
\code{urbinEla} uses a pseudo random number generator
to draw 1,000 values from a normal distribution
with the provided mean value and standard deviation of the covariate
and then imputes the covariance between the coefficients
of the linear and quadratic term of the covariate
as described in the previous paragraph.
The following command uses the simplified gradient
and---for the probit regression with the quadratic term---%
additionally an imputed value of the covariance between the coefficients
of the linear and quadratic term of the \code{age} variable
to calculate approximate standard errors of the semi-elasticities:
<<>>=
urbinEla( coef(estProbit), xMean, xPos = 3, model = "probit",
allCoefVcov = sqrt(diag(vcov(estProbit))) )
urbinEla( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
model = "probit", allCoefVcov = sqrt(diag(vcov(estProbitQ))),
xMeanSd = c( mean(Mroz87$age), sd(Mroz87$age) ) )
@
These standard errors are \emph{much} closer to the
standard errors based on the full variance-covariance matrices
than the na\"ive calculations
with the full gradients and replacing the missing covariances by zeros.
%========================== Interval coded covariates ==============================
\section{Semi-elasticities of interval-coded covariates}
\label{sec:elaint}
\subsection{Semi-elasticities}
In meta-analyses where the user is interested in comparing the semi-elasticities
of a certain continuous covariate across different studies,
studies that code the covariate of interest in intervals cause a serious problem,
as coefficients of interval-coded covariates cannot be compared to coefficients
or semi-elasticities of continuous covariates.
To overcome this problem, we suggest in this section a procedure to derive
a semi-elasticity of an interval-coded covariate.
A regression model with a binary dependent variable,
where the $k$th covariate is interval-coded can be specified as:
\begin{align}
\Pr( Y = 1 | X = x ) &= g \left( \beta_0 +
\sum_{j \in \{ 1, \ldots, K \} \setminus k } \beta_j x_j +
\sum_{m \in \{ 1, \ldots, M \} \setminus m^* } \delta_m D_m \right) , \\
D_m & =
\begin{cases}
1 & \text{ if } b_{m-1} < x_k \leq b_m \\
0 & \text{ otherwise}
\end{cases}
\quad \forall \; m = 1, \ldots , M,
\end{align}
where $g()$ is a generic link function that can take any form,
$x = ( x_1, \ldots , x_K )^{\top}$ is a vector of $K$ continuous covariates,
whereas the actual values of one of these covariates, $x_k$, are unobserved,
$D = ( D_1 , \ldots , D_M )^{\top}$ is a vector of $M$~dummy variables
that indicates in which intervals the values of covariate $x_k$ fall,
$b = ( b_0 , \ldots , b_M )^{\top}$ is a vector of the $M + 1$
boundaries of the $M$~intervals of covariate~$x_k$
with $b_0 < b_1 < \ldots < b_{M-1} < b_M$,
$m^* \in \{ 1 , \ldots , M \}$ is an arbitrary chosen interval
that is used as `base' interval in the regression,
and $\beta = ( \beta_0, \ldots, \beta_{k-1}, \beta_{k+1}, \ldots, \beta_K )^{\top}$ and
$\delta = ( \delta_1, \ldots, \delta_{m^* - 1}, \delta_{m^* + 1} ,
\ldots, \delta_M )^{\top}$
are vectors of $K$ and $M - 1$ unknown coefficients, respectively.
For convenience of further calculations,
we define the (non-estimated) coefficient for the `base' interval to be zero,
i.e., $\delta_{m^*} = 0$.
To derive the semi-elasticity of the `unknown' continuous $k$th covariate:
\begin{equation}
\epsilon_k \equiv \frac{\partial \Pr( Y = p | X = x )}{\partial x_k } \cdot x_k ,
\end{equation}
we calculate the effect of an increase of the $k$th covariate
above each inner boundary to the next higher interval
on the probability of $Y=1$, i.e.:
\begin{align}
e_{km} = \; &
\Pr( Y = 1 | b_m < x_k \leq b_{m+1} ) - \Pr( Y = 1 | b_{m-1} < x_k \leq b_m )\\
& \; \forall \; m = 1, \ldots, M-1
\nonumber
\end{align}
and the approximate proportions of observations
at which the $k$th covariate will increase above an inner boundary
if the $k$th covariate increases by one percent
around each of these boundaries
(i.e., the proportions of observations in the intervals
$\pm 0.5$\% around each inner boundary
assuming a uniform distribution of the values of the $k$th covariate
within each interval):
\begin{align}
p_{km} \approx \; &
0.005 \cdot b_m \frac{s_m}{b_m - b_{m-1}} +
0.005 \cdot b_m \frac{s_{m+1}}{b_{m+1} - b_m}
\; \forall \; m = 1, \ldots, M-1,
\end{align}
where $s_m$ is the proportion of observations
that are in the $m$th interval,
i.e., $b_{m-1} < x_k \leq b_m$.
Finally, we can calculate the approximate semi-elasticity by:
\begin{align}
\epsilon_k
& \approx 100 \cdot \sum_{m=1}^{M-1} e_{km} \cdot p_{km} \\
& \approx \sum_{m=1}^{M-1} \frac{e_{km} \cdot b_m}{2}
\left( \frac{s_m}{b_m - b_{m-1}} +
\frac{s_{m+1}}{b_{m+1} - b_m}
\right)\\
& \approx \sum_{m=1}^{M-1} e_{km} w_m
\label{eq:linElaIntAvg}\\
\text{with } w_m
& \equiv \frac{b_m}{2}
\left( \frac{s_m}{b_m - b_{m-1}} +
\frac{s_{m+1}}{b_{m+1} - b_m}
\right) \; \forall \; m = 1, \ldots, M
\end{align}
so that this semi-elasticity can be interpreted
in the same way as the semi-elasticity defined
in Section~\ref{sec:semela},
i.e., it indicates the approximate increase in the probability of $Y=1$
(in percentage points)
that is caused by a one percent increase of the covariate~$x_k$.
Table~\ref{tab:semiInt} presents the equations for calculating the semi-elasticities
of interval-coded covariates for all six estimation methods covered in this article.
\begin{sidewaystable}[hp]
\caption{Semi-elasticities of interval-coded covariates
for 6~different regression models}
\label{tab:semiInt}
\begin{tabular}{ p{5cm} p{15cm} }
\\[-1.8ex]\hline
\hline \\[-1.8ex]
Regression model & Semi-elasticity \\
\midrule \\[-1.8ex]
Linear probability model &
\begin{equation*}
\epsilon_k
\approx \sum_{m=1}^{M-1} \;
\left( \delta_{m+1} - \delta_m \right) w_m
\end{equation*} \\
Binary, bivariate / multivariate, and ordered probit regression &
\begin{equation*}
\begin{aligned}
\epsilon_k
& \approx \sum_{m=1}^{M-1} \;
\left( \Phi_{m+1} (\cdot) - \Phi_m (\cdot) \right) w_m \\
\text{with } \Phi_m (\cdot)
& \equiv \Phi \left( \beta_0 +
\sum_{j\in\{1, \ldots, K \} \setminus k} \beta_j x_j +
\delta_m \right) \; \forall \; m = 1, \ldots, M
\end{aligned}
\end{equation*}\\
Logistic regression &
\begin{equation*}
\begin{aligned}
\epsilon_k
& \approx \sum_{m=1}^{M-1}
\left( \frac{ \exp_{m+1} (\cdot) }{ 1 + \exp_{m+1}(\cdot) } -
\frac{ \exp_m (\cdot) }{ 1 + \exp_m (\cdot) } \right) w_m \\
\text{with } \exp_m (\cdot)
& \equiv \exp \left( \beta_0 +
\sum_{j \in \{1, \ldots, K \} \setminus k } \beta_j x_j +
\delta_m \right) \; \forall \; m = 1, \ldots, M
\end{aligned}
\end{equation*}\\
Multinomial logistic regression &
\begin{equation*}
\begin{aligned}
\epsilon_{k,\mathcal{P}} \approx \;
& \sum_{p \in \mathcal{P}} \; \epsilon_{k,p}^*\\
\text{with } \epsilon_{k,p}^* \approx \;
& \sum_{m=1}^{M-1}
\left( \frac{ \exp_{m+1,p} (\cdot) }{
\sum_{o=1}^P \exp_{m+1,o} (\cdot) } -
\frac{ \exp_{m,p} (\cdot) }{
\sum_{o=1}^P \exp_{m,o} (\cdot) } \right) w_m
\; \forall \; p = 1, \ldots, P \\
\text{and } \exp_{m,p} (\cdot) \equiv \;
& \exp \left( \beta_{0,p} +
\sum_{j \in \{1, \ldots, K \} \setminus k} \beta_{j,p} x_j +
\delta_{m,p} \right)
\; \forall \; m = 1, \ldots, M; p = 1, \ldots, P
\end{aligned}
\end{equation*}\\[1em]
\bottomrule
\end{tabular}\\[1ex]
Note:
\noteMLogitP
\end{sidewaystable}
To demonstrate how to calculate the semi-elasticity of the interval-coded variable
\code{age} with regard to a married woman's probability to participate in the labour force,
we estimate equation~(\ref{eq:esti}) as a logistic regression
with \code{age} as interval-coded covariate.
We create four intervals,
30--37, 38--44, 45--52, and 53--60~years,
and we use the third interval (45--52~years) as `base' interval
in the regression analysis.%
\footnote{%
The \R{} code for estimating this model
is available in Appendix Section~\ref{sec:RCodeEstLogitInt}.
}
The results of this estimation are presented in Table~\ref{tab:logitInt}.
<<estLogitInt, echo=FALSE, results='hide' >>=
estLogitInt <- glm( lfp ~ kids + age30.37 + age38.44 + age53.60 + educ,
family = binomial(link = "logit"), data = Mroz87 )
xMeanInt <- c( xMean[1:2], mean( Mroz87$age30.37 ),
mean( Mroz87$age38.44 ), mean( Mroz87$age53.60 ), xMean[4] )
@
<<tabEstLogitInt, echo=FALSE, results='asis' >>=
stargazer( estLogitInt,
title = "Logistic regression results with age as interval-coded covariate",
label = "tab:logitInt", header = FALSE,
digits = 2, intercept.bottom = FALSE )
@
Using the vector of coefficient estimates
that can be obtained from Table~\ref{tab:logitInt}
(\code{coef(estLogitInt)}) and
a vector with the sample means of the covariates \code{kids} and \code{educ}
and the proportions of observations in the three age intervals included in the regression
(\code{xMeantInt}),
one can calculate the semi-elasticity of the covariate \code{age}
using function \code{urbinElaInt}:
<<>>=
urbinElaInt( coef(estLogitInt), xMeanInt, xPos = c( 3, 4, 0, 5 ),
xBound = c( 30, 37.5, 44.5, 52.5, 60 ), model = "logit" )
@
Argument \code{xPos} indicates the positions of the four age intervals
(in ascending order)
in the vectors \code{coef(estLogitInt)} and \code{xMeantInt},
where a zero indicates the position of the reference interval
that was not included in the regression
and, thus, is not included in \code{coef(estLogitInt)} or \code{xMeantInt}.
Argument \code{xBound} indicates the five boundaries of the four intervals.
As the coefficients are derived from a logistic regression,
we set argument \code{model} equal to \code{"logit"}.
The semi-elasticity
based on the logistic regression with \code{age} as interval-coded covariate
indicates
that the probability that a woman is in the labour force decreases,
\textit{ceteris paribus}, by
\Sexpr{-round(urbinElaInt( coef(estLogitInt), xMeanInt, xPos = c( 3, 4, 0, 5 ), xBound = c( 30, 37.5, 44.5, 52.5, 60 ), model = "logit" )$semEla, 2 )}%
~percentage points if her age increases by one percent.
\subsection{Approximation of standard errors}
An approximate standard error of the semi-elasticity
of interval-coded covariates
can, again, be obtained by using the Delta method
(equation~\ref{eq:DeltaMethod}).
The gradient vectors of the semi-elasticities
with respect to the coefficients,
$\partial \epsilon_k / \partial ( \bm{\beta}^{\top} \bm{\delta}^{\top} )^{\top}$,
for the various regression models
are presented in Appendix Section~\ref{sec:gradSemElaInt}.
Argument \code{allCoefVcov} of function \code{urbinElaInt}
can be used to specify the variance-covariance matrix:
<<>>=
urbinElaInt( coef(estLogitInt), xMeanInt, xPos = c( 3, 4, 0, 5 ),
xBound = c( 30, 37.5, 44.5, 52.5, 60 ), model = "logit",
allCoefVcov = vcov(estLogitInt) )
@
As most studies do not report the variance-covariance matrix,
we repeat the above calculation
with providing only the standard errors
so that \code{urbinElaInt} sets all covariances to zero:
<<>>=
urbinElaInt( coef(estLogitInt), xMeanInt, xPos = c( 3, 4, 0, 5 ),
xBound = c( 30, 37.5, 44.5, 52.5, 60 ), model = "logit",
allCoefVcov = sqrt(diag(vcov(estLogitInt))) )
@
In this empirical example---as in most of our other empirical tests---%
setting the covariances to zero resulted in a slight overestimation
of the standard error
and we did not find a way to get better approximations
than with just setting the covariances to zero.
As setting the covariances to zero usually results
only in a slight overestimation of the standard errors,
we consider this approximation of the standard errors
(which are often anyway only used as weighting factors)
as generally suitable for meta-analyses.
%========================== Effect continuous covariates ==============================
\section{Effects of continuous covariates when they change between intervals}
\label{sec:effcont}
\subsection{Effect size}
In this sections, we consider the case
where the user wants to compare effects of an interval-coded covariate
on the probability of $Y = 1$.
We suggest a procedure that uses the results of studies
that use the covariate of interest in its continuous form
to calculate the effect of this covariate
when it switches from one reference interval to another interval.
We start out with a regression equation where the covariate of interest, $x_k$,
is included as a linear term:
\begin{equation}
Pr( Y = 1 | X = x ) = g\left( \beta_0 + \sum_{k=1}^K \beta_k x_k \right)
\end{equation}
We suggest to derive the (approximate) effects of $x_k$ on $Y$,
if this covariate changes between $M \geq 2$ discrete intervals,
e.g., from a `reference' interval~$r$ to an interval of interest~$l$,
by:
\begin{align}
E_{k,lr}
= \; & \Pr( Y = 1 | b_{l-1} < x_k \leq b_l, x_{-k} )
\label{eq:linEffectStart} \\
& - \Pr( Y = 1 | b_{r-1} < x_k \leq b_r, x_{-k} )
\nonumber \\
= \; & g \left( \beta_0 + \sum_{j \in \{ 1, \ldots, K \} \setminus k} \beta_j x_j
+ \beta_k E[ x_k | b_{l-1} < x_k \leq b_l ] \right) \\
& - g \left( \beta_0 - \sum_{j \in \{ 1, \ldots, K \} \setminus k} \beta_j x_j
+ \beta_k E[ x_k | b_{r-1} < x_k \leq b_r ] \right)
\nonumber \\[1ex]
= \; & g \left( \beta_0 + \sum_{j \in \{ 1, \ldots, K \} \setminus k} \beta_j x_j
+ \beta_k \bar{x}_{kl} \right) \label{eq:contEffect} \\
& - g \left( \beta_0 - \sum_{j \in \{ 1, \ldots, K \} \setminus k} \beta_j x_j
+ \beta_k \bar{x}_{kr} \right), \nonumber
\end{align}
where $x_{-k} = ( x_1 , \ldots , x_{k-1} , x_{k+1} , x_K )^{\top}$
is a vector of all covariates except for~$x_k$,
$b_0 < b_1 < \ldots < b_{M-1} < b_M$ are the boundaries of the intervals of covariate~$x_k$,
and
\begin{equation}
\bar{x}_{km} \equiv E[ x_k | b_{m-1} < x_k \leq b_m ]
\; \forall \; m = 1, \ldots , M
\end{equation}
are the expected values of covariate~$x_k$ within specific intervals.
If the expected values of covariate~$x_k$ for specific intervals are unknown,
it may be appropriate
to approximate them by the mid-points of the respective interval boundaries
(e.g., if the covariate~$x_k$ has approximately a uniform distribution
between the respective interval boundaries):
\begin{equation}
\bar{x}_{km} \approx \frac{ b_{m-1} + b_m }{2} \; \forall \; m = 1, \ldots , M.
\label{eq:linEffectXBar}
\end{equation}
If the model specification additionally includes a quadratic term
of the covariate~$k$,
e.g., $x_{k+1} = x_k^2$,
equations~(\ref{eq:linEffectStart}) to~(\ref{eq:contEffect}) change to:
\begin{align}
E_{k,lr}
= \; & \Pr( Y = 1 | b_{l-1} < x_k \leq b_l, x_{-k,k+1} )\\
& - \Pr( Y = 1 | b_{r-1} < x_k \leq b_r, x_{-k,k+1} )
\nonumber \\
= \; & g \bigg( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j \\
& + \beta_k E[ x_k | b_{l-1} < x_k \leq b_l ]
+ \beta_{k+1} E[ x_k^2 | b_{l-1} < x_k \leq b_l ] \bigg) \nonumber \\
& - g \bigg( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
\nonumber \\
& + \beta_k E[ x_k | b_{r-1} < x_k \leq b_r ]
+ \beta_{k+1} E[ x_k^2 | b_{r-1} < x_k \leq b_r ] \bigg)
\nonumber \\[1ex]
= \; & g \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \}} \beta_j x_j
+ \beta_k \bar{x}_{kl} + \beta_{k+1} \overline{x^2_{kl}} \right) \\
& - g \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \}} \beta_j x_j
+ \beta_k \bar{x}_{kr} + \beta_{k+1} \overline{x^2_{kr}} \right) , \nonumber
\label{eq:quadEffect}
\end{align}
with
\begin{equation}
\overline{x^2_{km}} \equiv E[ x_k^2 | b_{m-1} < x_k \leq b_m ]
\; \forall \; m = 1, \ldots , M.
\end{equation}
If $E[ x_k^2 | b_{m-1} < x_k \leq b_m ]$ is unknown,
it may be appropriate to approximate it by assuming
that covariate~$x_k$ has approximately a uniform distribution
between each pair of subsequent interval boundaries
so that its probability density function
between boundaries $b_{m-1}$ and $b_m$
is $1 / ( b_m - b_{m-1} )$:
\begin{align}
\overline{x^2_{km}}
& \approx \int_{b_{m-1}}^{b_m} x_k^2 \; \frac{1}{b_m - b_{m-1}} \; d \, x_k\\
& = \left. \frac{1}{3} \; x_k^3 \; \frac{1}{b_m - b_{m-1}}
\right|_{b_{m-1}}^{b_m} \\
& = \frac{1}{3} \; b_m^3 \; \frac{1}{b_m - b_{m-1}}
- \frac{1}{3} \; b_{m-1}^3 \; \frac{1}{b_m - b_{m-1}} \\
& = \frac{ b_m^3 - b_{m-1}^3 }{ 3 \left( b_m - b_{m-1} \right)}
\; \forall \; m = 1, \ldots, M.
\label{eq:linEffectXSquaredBar}
\end{align}
Table~\ref{tab:effCont} presents the equations for calculating the effect sizes
of continuous covariates when they change between intervals
for all six estimation methods covered in this article.
\begin{sidewaystable}[hp]
\caption{Effects of continuous covariates when they change between intervals
for 6~different regression models}
\label{tab:effCont}
\begin{tabular}{ p{5cm} p{15cm} }
\\[-1.8ex]\hline
\hline \\[-1.8ex]
Regression model & Effect size \\
\midrule \\[-1.8ex]
Linear probability model &
\begin{equation*}
E_{k,lr} \approx
\beta_k \left( \bar{x}_{kl} - \bar{x}_{kr} \right)
+ \beta_{k+1} \left( \overline{x^2_{kl}} - \overline{x^2_{kr}} \right)
\label{eq:semELPM}
\end{equation*} \\
Binary, bivariate / multivariate, and ordered probit regression &
\begin{equation*}
\begin{aligned}
E_{k,lr} \approx \;
& \Phi\left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kl} + \beta_{k+1} \overline{x^2_{kl}} \right)\\
& - \Phi\left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kr} + \beta_{k+1} \overline{x^2_{kr}} \right)
\end{aligned}
\end{equation*}\\
Logistic regression &
\begin{equation*}
\begin{aligned}
E_{k,lr} \approx \;
& \frac{ \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kl} + \beta_{k+1} \overline{x^2_{kl}} \right)}
{ 1 + \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kl} + \beta_{k+1} \overline{x^2_{kl}} \right)}\\
& - \frac{ \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kr} + \beta_{k+1} \overline{x^2_{kr}} \right)}
{ 1 + \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kr} + \beta_{k+1} \overline{x^2_{kr}} \right)}
\end{aligned}
\end{equation*}\\
Multinomial logistic regression &
\begin{equation*}
\begin{aligned}
E_{k,lr,\mathcal{P}} \approx \;
& \sum_{p \in \mathcal{P}} E_{k, lr, p}^*\\
\text{with } E_{k, lr, p}^* \equiv \;
& \frac{ \exp \left( \beta_{0,p}
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,p} x_j
+ \beta_{k,p} \bar{x}_{kl} + \beta_{k+1,p} \overline{x^2_{kl}} \right)}
{ \sum_{o=1}^P
\exp \left( \beta_{0,o}
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,o} x_j
+ \beta_{k,o} \bar{x}_{kl} + \beta_{k+1,o} \overline{x^2_{kl}} \right)} \\
& - \frac{ \exp \left( \beta_{0,p}
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,p} x_j
+ \beta_{k,p} \bar{x}_{kr} + \beta_{k+1,p} \overline{x^2_{kr}} \right)}
{ \sum_{o=1}^P
\exp \left( \beta_{0,o}
+ \sum_{j \in \{ 1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,o} x_j
+ \beta_{k,o} \bar{x}_{kr} + \beta_{k+1,o} \overline{x^2_{kr}} \right)}
\end{aligned}
\end{equation*} \\[1em]
\bottomrule
\end{tabular}\\[1ex]
Note:
If the regression analysis
does not include a quadratic term of the covariate of interest,
$\beta_{k+1}$ is simply the coefficient of the covariate~$x_{k+1}$
so that both $\overline{x^2_{kl}}$ and $\overline{x^2_{kr}}$
in the above equations must be replaced by~$x_{k+1}$.
\noteMLogitP
\end{sidewaystable}
The effect of a continuous covariate
when it changes between intervals
can be calculated with package \pkg{urbin}
by using function \code{urbinEffInt}.
In our example,
we use the results of the probit regression model
with variable \code{age} as a linear covariate
as well as with \code{age} as a linear and a quadratic covariate
that we already used as example in Section~\ref{sec:semela}
with estimation results presented in Table~\ref{tab:probitSum}.
Based on these estimation results,
we calculate the effect of covariate \code{age}
on the probability of women's participation in the labour force
as if \code{age} was an interval-coded covariate
and changes from the 30--44~years (reference) interval
to the 53--60~years interval.
We do this both for the model with \code{age} as linear covariate:
<<>>=
urbinEffInt( coef(estProbit), xMean, xPos = 3,
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit" )
@
and for the model with \code{age} as a linear and quadratic covariate:
<<>>=
urbinEffInt( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit" )
@
The results based on the two estimated models indicate
that the probability that a woman is in the labour force
is, \textit{ceteris paribus},
\Sexpr{round(-100*urbinEffInt( coef(estProbit), xMean, xPos = 3, refBound = c( 30, 44 ), intBound = c( 53, 60 ), model = "probit" )$effect)}%
~percentage points or
\Sexpr{round(-100*urbinEffInt( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ), refBound = c( 30, 44 ), intBound = c( 53, 60 ), model = "probit" )$effect)}%
~percentage points, respectively, lower for women aged 53--60~years
than for women aged 30--44~years.
\subsection{Approximation of standard errors}
As for the semi-elasticities,
an approximate standard error of the effect
of interval-coded covariates
can be obtained by using the Delta method
(equation~\ref{eq:DeltaMethod}).
Appendix Section~\ref{sec:gradEffectInt} presents
the gradient vectors of the effects
with respect to the coefficients,
$\partial E_{k,lr} / \partial \bm{\beta}$,
for the various regression models.
Argument \code{allCoefVcov} of function \code{urbinEffInt}
can be used to specify the variance-covariance matrix:
<<>>=
urbinEffInt( coef(estProbit), xMean, xPos = 3,
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit", allCoefVcov = vcov(estProbit) )
urbinEffInt( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit", allCoefVcov = vcov(estProbitQ) )
@
Given that most studies only report standard errors
rather than the (full) variance-covariance matrix,
we repeat the above calculations
with providing only the standard errors
so that \code{urbinEffInt} sets all covariances to zero:
<<>>=
urbinEffInt( coef(estProbit), xMean, xPos = 3,
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit", allCoefVcov = sqrt(diag(vcov(estProbit))) )
urbinEffInt( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit", allCoefVcov = sqrt(diag(vcov(estProbitQ))) )
@
While replacing the (frequently unknown) covariances by zeros
usually has only a minor effect on the standard error
when the model has only a linear term of the covariate of interest,
% (and this has no effect at all in case of a linear probability model
% as can be seen from the results above),
the standard errors based on models
with linear and quadratic terms of the covariate of interest
are usually largely upward-biased
if the covariances are all set to zeros.
However, approximating the covariance between the coefficient of the linear term
and the coefficient of the quadratic term
as explained in Section~\ref{sec:semElaSE}
usually gives sufficiently precise approximations of the standard error.
Function~\code{urbinEffInt} applies this procedure,
if the user provides the mean value and the standard deviation
of the covariate of interest through argument \code{xMeanSd}:
<<>>=
urbinEffInt( coef(estProbitQ), xMeanQ, xPos = c( 3, 4 ),
refBound = c( 30, 44 ), intBound = c( 53, 60 ),
model = "probit", allCoefVcov = sqrt(diag(vcov(estProbitQ))),
xMeanSd = c( mean( Mroz87$age ), sd( Mroz87$age ) ) )
@
\section{Grouping and re-basing effects of categorical and interval-coded covariates}
\label{sec:lingroup}
\subsection{Effect size}
In cases where the user is interested in comparing effects of categorical
or interval-coded covariates on a binary dependent variable,
the user will frequently encounter studies,
where the encoding of the covariate of interest
differs between studies,
e.g., the studies use different reference categories and/or
different categorisations.%
\footnote{%
In order to simplify the notation,
we use the term `categorical variables' throughout this section
although all derivations in this section
not only apply to (unordered or ordered) categorical variables
but equally apply to interval-coded variables.
}
In this section, we suggest an approach to obtain comparable effect sizes
by streamlining the categories
and unifying the reference category.
We consider a regression model:
\begin{align}
Pr( Y = 1 | X = x ) & = g\left( \beta_0 + \sum_{j \in \{ 1, \ldots , K \} \setminus k}
\beta_j x_j
+ \sum_{m \in \{ 1, \ldots , M \} \setminus m^* } \delta_m D_m \right),\\
D_m & =
\begin{cases}
1 & \text{ if } x_k \in c_m \\
0 & \text{ otherwise}
\end{cases}
\quad \forall \; m = 1, \ldots , M,
\end{align}
where $g()$ is again a generic link function
and the covariate of interest, $x_k$, is a categorical variable
with $M$ mutually exclusive categories $c_1, \ldots, c_M$
with $c_m \cap c_l = \emptyset \; \forall \; m \neq l$,
category $c_{m^*}$ is used as reference category,
and all other variables and coefficients are defined as above.
For notational simplification of the following derivations,
we define the (non-estimated) coefficient of the reference category to be zero,
i.e., $\delta_{m^*} \equiv 0$.
We want to obtain the effect of a change of covariate~$x_k$
from a reference category~$c_r^*$
to a category of interest~$c_l^*$:
\begin{align}
E_{k,lr}
& = \Pr( Y = 1 | x_k \in c_l^* ) - \Pr( Y = 1 | x_k \in c_r^* ),
\label{eq:effCat}
\end{align}
where categories~$c_r^*$ and/or~$c_l^*$
may comprise multiple original categories $c_1, \ldots, c_M$.
Vectors $v_r = ( v_{r1} , \ldots , v_{rM} )^{\top}$ and
$v_l = ( v_{l1} , \ldots , v_{lM} )^{\top}$
indicate,
which of the original categories $c_1, \ldots, c_M$
are included in categories~$c_r^*$ and~$c_l^*$,
respectively:
\begin{align}
v_{nm} & =
\begin{cases}
1 & \text{ if } c_m \in c_n^* \\
0 & \text{ if } c_m \notin c_n^*
\end{cases}
\forall \; m = 1, \ldots, M ; n \in \{ r , l \}
\end{align}
In the following,
we derive the effect of a change of covariate~$x_k$
from a reference category~$c_r^*$
to a category of interest~$c_l^*$,
$E_{k,lr}$ as defined in equation~(\ref{eq:effCat}):
\begin{align}
E_{k,lr}
= & \Pr( Y = 1 | x_k \in c_l^* ) - \Pr( Y = 1 | x_k \in c_r^* ) \\
= & g\left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m E[ D_m | x_k \in c_l^* ] \right) \\
& - g\left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^{M} \delta_m E[ D_m | x_k \in c_r^* ] \right)
\nonumber \\
= & g \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{ml} \right) \\
& - g \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^{M} \delta_m D_{mr} \right)
\nonumber
\end{align}
with
\begin{align}
D_{mn} & \equiv E[ D_m | x_k \in c_n^* ] \\
& = \frac{ P( D_m = 1 \cap x_k \in c_n^* ) }{ P( x_k \in c_n^* ) } \\
& = \frac{ E[ D_m ] \; v_{nm}}{ P( x_k \in c_n^* ) } \\
& = \frac{ s_m v_{nm} }{ \sum_{k=1}^M s_k v_{nk} }
\label{eq:linEffGrD}\\
& \qquad \forall \; m = 1, \ldots, M ; n \in \{ r , l \},
\nonumber
\end{align}
where $s_m = E[ D_m ] \; \forall \; m = 1, \ldots , M$
is the share of observations
with covariate~$x_k$ being in category~$c_m$.
Table~\ref{tab:effGroup} presents the equations for
grouping and re-basing effects of categorical and interval-coded covariates
for all six estimation methods covered in this article.
\begin{sidewaystable}[hp]
\caption{Grouping and re-basing effects of categorical and interval-coded covariates
for 6~different regression models}
\label{tab:effGroup}
\begin{tabular}{ p{5cm} p{15cm} }
\\[-1.8ex]\hline
\hline \\[-1.8ex]
Regression model & Effect size \\
\midrule \\[-1.8ex]
Linear probability model &
\begin{equation*}
E_{k,lr}
= \sum_{m=1}^{M} \delta_m \left( D_{ml} - D_{mr} \right)
\end{equation*} \\
Binary, bivariate / multivariate, and ordered probit regression &
\begin{equation*}
\begin{aligned}
E_{k,lr}
= \Phi \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{ml} \right)
- \Phi \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^{M} \delta_m D_{mr} \right)
\end{aligned}
\end{equation*}\\
Logistic regression &
\begin{equation*}
\begin{aligned}
E_{k,lr}
= \; & \frac{ \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{ml} \right)}
{ 1 + \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{ml} \right)}\\
& - \frac{ \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^{M} \delta_m D_{mr} \right)}
{ 1 + \exp \left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{mr} \right)}
\end{aligned}
\end{equation*}\\
Multinomial logistic regression &
\begin{equation*}
\begin{aligned}
E_{k, lr, \mathcal{P}} = \;
& \sum_{p \in \mathcal{P}} E_{k, lr, p}^* \\
E_{k, lr, p}^* = \;
& \frac{ \exp \left( \beta_{0,p}
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_{j,p} x_j
+ \sum_{m=1}^M \delta_{m,p} D_{ml} \right)}
{ \sum_{o=1}^P
\exp \left( \beta_{0,o}
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_{j,o} x_j
+ \sum_{m=1}^M \delta_{m,o} D_{ml} \right)} \\
& - \frac{ \exp \left( \beta_{0,p}
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_{j,p} x_j
+ \sum_{m=1}^{M} \delta_{m,p} D_{mr} \right)}
{ \sum_{o=1}^P
\exp \left( \beta_{0,o}
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_{j,o} x_j
+ \sum_{m=1}^M \delta_{m,o} D_{mr} \right)}
\end{aligned}
\end{equation*} \\[1em]
\bottomrule
\end{tabular}\\[1ex]
Note:
\noteMLogitP
\end{sidewaystable}
To demonstrate how to group and re-base a categorical covariate,
we use the results of the logistic regression model
with \code{age} as interval-coded covariate
that we already used as example in Section~\ref{sec:elaint}
with estimation results presented in Table~\ref{tab:logitInt}.
In this estimation, covariate \code{age} is coded as four intervals:
30--37~years, 38--44~years, 45--52~years, and 53--60~years,
where the interval 45--52~years is used as `base' interval.
In our example,
we apply function \code{urbinEffCat}
to group and re-base the categories to calculate
the effect of \code{age} changing from the 30--44~years (reference) interval
to the 53--60~years interval:
<<>>=
urbinEffCat( coef(estLogitInt), xMeanInt, xPos = c( 3:5 ),
xGroups = c( -1, -1, 1, 0 ), model = "logit" )
@
Argument \code{xPos} indicates the positions of the
categories of variable \code{age} in the coefficient vector and in the vector of mean values,
argument \code{xGroups} indicates how the four original categories
should be grouped and re-based,
and all other arguments are defined as explained for the other functions
of the \pkg{urbin} package.
Argument \code{xGroups} must have one element for each category
that was used in the estimation,
where the categories are in the same order as indicated by argument \code{xPos}
and the last element is the `base' category,
i.e., in our case, the elements of argument \code{xGroups}
must correspond to the order:
30--37~years (first element of \code{xPos}),
38--44~years (second element of \code{xPos}),
53--60~years (third element of \code{xPos}),
and 45--52~years (reference category).
Each element of argument \code{xGroups} must be
a $-1$ (indicating that the category should belong to the new reference category),
a $1$ (indicating that the category should belong to the new category of interest),
or a $0$ (indicating that the category should neither belong
to the new reference category nor to the new category of interest).
As the new reference category comprises
both the 30--37~years interval and the 38--44~years interval,
the values of the first two elements of argument \code{xGroups} must be~$-1$.
As the new category of interest is the 53--60~years interval,
the value of the third element of argument \code{xGroups} must be~$1$.
As the old reference interval, 45--52~years,
is neither in the new reference category nor in the new category of interest,
the value of the fourth element of argument \code{xGroups} must be~$0$.
The calculated effect size indicates
that the probability that a woman is in the labour force
is, \textit{ceteris paribus},
\Sexpr{round(-100*urbinEffCat( coef(estLogitInt), xMeanInt, xPos = c( 3:5 ), xGroups = c( -1, -1, 1, 0 ), model = "logit" )$effect)}%
~percentage points lower for women aged 53--60~years
than for women aged 30--44~years.
\subsection{Approximation of standard errors}
An approximate standard error of the effect of a grouped and re-based covariate
can, again, be obtained by using the Delta method
(equation~\ref{eq:DeltaMethod}).
Appendix Section~\ref{sec:gradEffectCat} presents
the gradient vectors of the effects
with respect to the coefficients,
$\partial E_{k,lr} / \partial ( \bm{\beta}^{\top} \bm{\delta}^{\top} )^{\top}$,
for the various regression models.
Argument \code{allCoefVcov} of function \code{urbinEffCat}
can be used to specify the variance-covariance matrix
of the estimated coefficients:
<<>>=
urbinEffCat( coef(estLogitInt), xMeanInt, c( 3:5 ),
c( -1, -1, 1, 0 ), vcov(estLogitInt), model = "logit" )
@
As most studies do not report the variance-covariance matrix,
we repeat the previous calculation
with providing only the standard errors
so that \code{urbinEffCat} sets all covariances to zero:
<<>>=
urbinEffCat( coef(estLogitInt), xMeanInt, c( 3:5 ),
c( -1, -1, 1, 0 ), sqrt(diag(vcov(estLogitInt))),
model = "logit" )
@
Similarly to Section~\ref{sec:elaint},
setting the covariances to zero
usually results in a slight overestimation of the standard errors.
As these overestimations are usually small
and the standard errors are often anyway only used as weighting factors,
we consider this approximation of the standard errors
to be generally suitable for meta-analyses.
\section{Non-binary categorical dependent variables}
\label{sec:non-binaryDepVar}
As explained in Section~\ref{sec:method},
it can be possible
to make results of studies with non-binary categorical dependent variables
comparable to results of studies with binary dependent variables,
if the categories of the (non-binary) dependent variable
can be grouped into two groups
that correspond to the two outcomes of the binary dependent variable
in the other studies.
In order to demonstrate this,
we use an ordered probit regression with \code{age}
as linear and quadratic covariate
and a multinomial logistic regression
with \code{age} as interval-coded covariate
as examples.%
\footnote{%
The \R{} code for estimating these two models
is available in Appendix Sections~\ref{sec:RCodeEstOProbit}
and~\ref{sec:RCodeEstMLogitInt}, respectively.
}
The estimation results of these two models are presented
in Tables~\ref{tab:estOProbit} and~\ref{tab:estMLogitInt},
respectively.
<<estOProbit,echo=FALSE, results='hide'>>=
library( "MASS" )
estOProbitQ <- polr( lfp3 ~ kids + age + I(age^2) + educ,
data = Mroz87, method = "probit", Hess = TRUE )
xMeanOProbit <- c( xMeanQ, -1 )
@
<<tabEstOProbit,echo=FALSE, results='asis'>>=
stargazer( estOProbitQ,
title = "Ordered probit regression results with age as linear and quadratic covariate",
label = "tab:estOProbit", header = FALSE,
ord.intercepts = TRUE, digits = 2 )
@
<<estMLogitInt,echo=FALSE, results='hide', message = FALSE>>=
library( "mlogit" )
estMLogitInt <- mlogit(
lfp3 ~ 0 | kids + age30.37 + age38.44 + age53.60 + educ,
data = Mroz87, reflevel = "no", shape = "wide" )
@
<<tabEstMLogit,echo=FALSE, results='asis'>>=
stargazer( estMLogitInt,
title = "Multinomial logistic regression results with age as interval-coded covariate",
label = "tab:estMLogitInt", header = FALSE,
digits = 2 )
@
We combine the two outcomes `part-time labour force participation' and
`full-time labour force participation'
to one joint outcome category
so that we obtain a binary outcome:
`no labour force participation' and
(part-time or full-time) 'labour force participation'.
For ordered probit models,
the negative value of the break point
that separates the two groups of categories
corresponds to the intercept
of a binary probit model (see Section~\ref{sec:method}).
Hence, in our example, the relevant break point is the one
between the `no labour force participation' category
and the `part-time labour force participation' category,
which has an estimated value of
\Sexpr{round(coef( summary( estOProbitQ ) )["no|part",1], 2)}
(see Table~\ref{tab:estOProbit}).
When applying one of the functions of the \pkg{urbin} package
to ordered probit models,
argument \code{iPos} must indicate
the position of this break point
in the vector of coefficients,
while all other break points must be ignored.
The element in the vector of the values of the covariates
that corresponds to the relevant break point
(as indicated by argument \code{iPos})
must be minus one,
in order to take into account
that the intercept of a corresponding binary probit model
must be replaced by the \emph{negative} value of the relevant break point
of an ordered probit model.
We set argument \code{model} to \code{"oprobit"}
to indicate an ordered probit model,
while all other arguments are used as explained above:
<<>>=
urbinEla( coef(summary(estOProbitQ))[-6,1], c( xMeanQ[-1], -1 ),
xPos = c( 2, 3 ), iPos = 5, model = "oprobit",
vcov(estOProbitQ)[-6,-6] )
@
The calculated semi-elasticity indicates
that the probability that a woman is at least part-time in the labour force
decreases,
\textit{ceteris paribus}, by
\Sexpr{-round(urbinEla( coef(summary(estOProbitQ))[-6,1], c( xMeanQ[-1], -1 ), xPos = c( 2, 3 ), iPos = 5, model = "oprobit" )$semEla, 2 )}%
~percentage points if her age increases by one percent.
In the multivariate logistic regression,
`no labour force participation' is used as the reference category
of the dependent variables,
while `full-time labour force participation' and
`part-time labour force participation' are used
as first alternative category and second alternative category,
respectively (see Table~\ref{tab:estMLogitInt}).
When applying one of the functions of the \pkg{urbin} package
to a multinomial logistic regression,
argument \code{yCat} must indicate the categories of the dependent variable $\mathcal{P}$
that correspond to a binary outcome of one.
All other categories are considered to correspond to a binary outcome of zero.
In argument \code{yCat}, a zero indicates the reference category,
while a one, two, or three, etc.\ indicates the first, second, or third, etc.\
alternative category, respectively.
As the first and second alternative categories comprise
the binary outcome of one in our example,
i.e., $\mathcal{P} = \{ 1, 2 \}$,
argument \code{yCat} must be a vector with two values: one and two.
We set argument \code{model} to \code{"mlogit"}
to indicate a multinomial logistic regression,
while all other arguments are used as explained above:
<<>>=
coefPermuteInt <- c( seq( 1, 11, 2 ), seq( 2, 12, 2 ) )
urbinElaInt( coef(estMLogitInt)[coefPermuteInt], xMeanInt,
c( 3, 4, 0, 5 ), c( 30, 37.5, 44.5, 52.5, 60 ), model = "mlogit",
vcov(estMLogitInt)[coefPermuteInt,coefPermuteInt],
yCat = c( 1, 2 ) )
@
As the functions in package \pkg{urbin} expect
that the coefficients of multinomial logistic regressions
are grouped for each category of the dependent variable
(i.e., $\beta_{0,1}, \ldots, \beta_{K,1}, \beta_{0,2}, \ldots, \beta_{K,2},
\ldots, \beta_{0,P}, \ldots, \beta_{K,P}$),
while the coefficients of models estimated by \code{mlogit}
are grouped for each covariate
(i.e., $\beta_{0,1}, \ldots, \beta_{0,P}, \beta_{1,1}, \ldots, \beta_{1,P},
\ldots, \beta_{K,1}, \ldots, \beta_{K,P}$),
we created a vector \code{coefPermuteInt}
that reorders the coefficients and their variances and covariances
so that they are ordered as expected by package \pkg{urbin}.
The semi-elasticity indicates
that the probability
that a woman is either part-time or full-time in the labour force
decreases, \textit{ceteris paribus}, by
\Sexpr{-round(urbinElaInt( coef(estMLogitInt)[coefPermuteInt], xMeanInt, c( 3, 4, 0, 5 ), c( 30, 37.5, 44.5, 52.5, 60 ), model = "mlogit", yCat = c( 1, 2 ) )$semEla, 2 )}%
~percentage points if her age increases by one percent.
\section{Conclusion}
The direct comparison of coefficients from regression analyses from different studies is often
meaningless because the studies use different estimation methods
or different units of measurements or different encodings
of the variables of interest.
In this article, we propose straightforward and easy-to-implement approaches to
unify results from regression analyses with binary dependent variables
or categorical dependent variables that can be transformed to binary variables.
We have implemented all suggested approaches in the \R{} package \pkg{urbin}.
This article uses this package to demonstrate
how regression results from differently specified regression analyses
can be unified by calculating semi-elasticities
of continuous and interval-coded covariates,
by calculating effects of continuous covariates
when they change between intervals,
and by grouping and re-basing effects of categorical
and interval-coded covariates.
We show how to obtain valid approximations for
the calculated standard errors of the semi-elasticities and effect sizes
without information about the variance-covariance matrix of the coefficients,
e.g., for cases where the user wants to use the standard errors as
weighting factors in a meta-analysis.
\clearpage
\bibliographystyle{ama2}
\bibliography{references}
\clearpage
%===================================== Appendix ======================================
\appendix
\section{Gradients for calculating approximate standard errors}
%+++++++++++++++++++++++++++++++++++ semEla ++++++++++++++++++++++++++++++++++
\subsection{Gradients of semi-elasticities of continuous covariates}
\label{sec:gradSemEla}
%================================= semEla: LPM ===============================
\subsubsection{Linear probability model}
\label{sec:gradSemElaLpm}
If the regression equation includes only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_j} &= 0
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial \epsilon_k}{\partial \beta_k} &= x_k.
\end{align}
If the regression equation additionally includes a quadratic term of the covariate of interest,
there is one additional gradient:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_{k+1}} &= 2 \; x_k^2.
\end{align}
%============================= semEla: Probit ================================
\subsubsection{Probit regression}
If the regression equation includes only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_j} &=
- \bm{X}'\bm{\beta} \; \epsilon_k \; x_j
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial \epsilon_k}{\partial \beta_k} &=
- \bm{X}'\bm{\beta} \; \epsilon_k \; x_k
+ \phi( \bm{X}'\bm{\beta} ) \; x_k
\end{align}
with $x_0 \equiv 1$.
If the regression equation additionally includes a quadratic term of the covariate of interest,
there is one additional gradient:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_{k+1}} &=
- \bm{X}'\bm{\beta} \; \epsilon_k \; x_k^2
+ 2 \; \phi( \bm{X}'\bm{\beta} ) \; x_k^2.
\end{align}
%================================ semEla: Logit ===============================
\subsubsection{Logistic regression}
If the regression equation includes only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_j} &=
\left( 1 -
\frac{ 2 \; \exp \left( \bm{X}'\bm{\beta} \right) }
{ 1 + \exp \left( \bm{X}'\bm{\beta} \right) } \right)
\epsilon_k \; x_j \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial \epsilon_k}{\partial \beta_k} &=
\left( 1 -
\frac{ 2 \; \exp \left( \bm{X}'\bm{\beta} \right) }
{ 1 + \exp \left( \bm{X}'\bm{\beta} \right) } \right)
\epsilon_k \; x_k
+ \frac{ \exp \left( \bm{X}'\bm{\beta} \right) }
{ \left( 1 + \exp \left( \bm{X}'\bm{\beta} \right) \right)^2 } \; x_k
\end{align}
with $x_0 \equiv 1$ (see also \cite{Train2002}).
If the regression equation additionally includes a quadratic term of the covariate of interest,
there is one additional gradient:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_{k+1}} &=
\left( 1 -
\frac{ 2 \; \exp \left( \bm{X}'\bm{\beta} \right) }
{ 1 + \exp \left( \bm{X}'\bm{\beta} \right) } \right)
\epsilon_k \; x_k^2
+ 2 \; \frac{ \exp \left( \bm{X}'\bm{\beta} \right) }
{ \left( 1 + \exp \left( \bm{X}'\bm{\beta} \right) \right)^2 } x_k^2.
\end{align}
%================================== semEla: MNL ==============================
\subsubsection{Multinomial logistic regression}
If the regression equation includes only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \beta_{j,o} }
\; \forall \; j = 0, \ldots, K;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{j,o}}
= \; & \left( - \pi_p \epsilon_{k,o}^* - \pi_o \epsilon_{k,p}^*
+ \Delta_{o,p} \epsilon_{k,p}^* \right) x_j \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \nonumber \\
\frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{k,o}}
= \; & \left( - \pi_p \epsilon_{k,o}^* - \pi_o \epsilon_{k,p}^*
- \pi_p \pi_o
+ \Delta_{o,p} \left( \pi_p + \epsilon_{k,p}^* \right)
\right) x_k\\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^* \nonumber,
\end{align}
$x_0 \equiv 1$, and $\Delta_{o,p}$ denoting Kronecker's Delta
with $\Delta_{o,p} = 1 \; \forall \; o = p$
and $\Delta_{o,p} = 0 \; \forall \; o \neq p$.
If the regression equation additionally includes a quadratic term of the covariate of interest,
there are $P-1$ additional gradients:
\begin{align}
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \beta_{k+1,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \beta_{k+1,o} }
\; \forall \; o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{k+1,o}}
= \; & \left( - \pi_p \epsilon_{k,o}^* - \pi_o \epsilon_{k,p}^*
- 2 \pi_p \pi_o
+ \Delta_{o,p} \left( 2 \pi_p + \epsilon_{k,p}^* \right)
\right) x_k^2\\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^* \nonumber.
\end{align}
%+++++++++++++++++++++++++++++++ semEla (simplified) +++++++++++++++++++++++++
\subsection{Simplified gradients of semi-elasticities of continuous covariates}
\label{sec:gradSemElaSimple}
%========================== semEla (simplified): LPM =========================
\subsubsection{Linear probability model}
As almost all elements of the gradient vector are zero
(see section~\ref{sec:gradSemElaLpm}),
almost all off-diagonal elements of the variance-covariance matrix
of the estimated coefficients are anyway ignored
when the Delta method is applied to calculate the approximate standard error
of the semi-elasticity.
Therefore, we do not need to obtain `simplified' gradients
in order to avoid biases due to missing information
about the off-diagonal elements of the variance-covariance matrix.
%======================== semEla (simplified): Probit ========================
\subsubsection{Probit regression}
In order to improve the approximation of the standard errors
when the off-diagonal elements of the variance-covariance matrix
of the estimated coefficients are unknown,
we simplify the derivation of the gradients by ignoring
that the `weighting factor' $\phi(\cdot)$ in the equation
for calculating the semi-elasticities (see Table~\ref{tab:semiE})
depends on the coefficients.
If the regression equation includes only a linear term of the covariate of interest,
the `simplified' gradients are:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_j} &=
0 \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial \epsilon_k}{\partial \beta_k} &=
\phi( \bm{X}'\bm{\beta} ) \; x_k.
\end{align}
If the regression equation additionally includes a quadratic term of the covariate of interest,
there is one additional `simplified' gradient:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_{k+1}} &=
2 \; \phi( \bm{X}'\bm{\beta} ) \; x_k^2.
\end{align}
%=========================== semEla (simplified): Logit ===============================
\subsubsection{Logistic regression}
In order to improve the approximation of the standard errors
when the off-diagonal elements of the variance-covariance matrix
of the estimated coefficients are unknown,
we simplify the derivation of the gradients by ignoring
that the `weighting factor' $\exp(\cdot) / ( 1 - \exp(\cdot) )^2$ in the equation
for calculating the semi-elasticities (see Table~\ref{tab:semiE})
depends on the coefficients.
If the regression equation includes only a linear term of the covariate of interest,
the `simplified' gradients are:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_j} &=
0 \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial \epsilon_k}{\partial \beta_k} &=
\frac{ \exp \left( \bm{X}'\bm{\beta} \right) }
{ \left( 1 + \exp \left( \bm{X}'\bm{\beta} \right) \right)^2 } \; x_k
\end{align}
If the regression equation additionally includes a quadratic term of the covariate of interest,
there is one additional `simplified' gradient:
\begin{align}
\frac{\partial \epsilon_k}{\partial \beta_{k+1}} &=
2 \; \frac{ \exp \left( \bm{X}'\bm{\beta} \right) }
{ \left( 1 + \exp \left( \bm{X}'\bm{\beta} \right) \right)^2 } x_k^2.
\end{align}
%============================= semEla (simplified): MNL ======================
\subsubsection{Multinomial logistic regression}
In order to improve the approximation of the standard errors
when the off-diagonal elements of the variance-covariance matrix
of the estimated coefficients are unknown,
we simplify the derivation of the gradients by ignoring
that the `weighting factors' $\pi_p ; p = \{ 1, \ldots , P \}$
and $\pi_o ; o = \{ 1, \ldots , P \}$ in the equation
for calculating the semi-elasticities (see Table~\ref{tab:semiE})
depend on the coefficients.
If the regression equation includes only a linear term of the covariate of interest,
the `simplified' gradients are:
\begin{align}
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \beta_{j,o} }
\; \forall \; j = 0, \ldots, K;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{j,o}}
= \; & 0 \; \forall \; j \in \{ 0, \ldots, K \} \setminus k; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \\
\frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{k,o}}
= \; & \left( - \pi_p \pi_o + \Delta_{o,p} \pi_p \right) x_k
\; \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*,
\end{align}
and $\Delta_{o,p}$ denoting Kronecker's Delta
with $\Delta_{o,p} = 1 \; \forall \; o = p$
and $\Delta_{o,p} = 0 \; \forall \; o \neq p$.
If the regression equation additionally includes a quadratic term of the covariate of interest,
there are $P-1$ additional gradients:
\begin{align}
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \beta_{k+1,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \beta_{k+1,o} }
\; \forall \; o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{ \partial \epsilon_{k,p}^*}{\partial \beta_{k+1,o}}
= \; & \left( - 2 \pi_p \pi_o + 2 \Delta_{o,p} \pi_p \right) x_k^2 \\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*. \nonumber
\end{align}
%+++++++++++++++++++++++++++++++++ semElaInt +++++++++++++++++++++++++++++++++
\subsection{Gradients of semi-elasticities of interval-coded covariates}
\label{sec:gradSemElaInt}
%================================ semElaInt: LPM ==============================
\subsubsection{Linear probability model}
The gradients are:
\begin{align}
\frac{ \partial \epsilon_k }{ \partial \beta_j }
= \; & 0 \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{ \partial \epsilon_k }{ \partial \delta_1 }
= \; & - w_1 \\
\frac{ \partial \epsilon_k }{ \partial \delta_m }
= \; & w_{m-1} - w_m
\; \forall \; m \in \{2, \ldots, M-1 \} \setminus m^* \\
\frac{ \partial \epsilon_k }{ \partial \delta_M }
= \; & w_{M-1}
\end{align}
%=========================== semElaInt: Probit ===============================
\subsubsection{Probit regression}
The gradients are:
\begin{align}
\frac{ \partial \epsilon_k }{ \partial \beta_j }
= \; & x_j \sum_{m=1}^{M-1} \left( \phi_{m+1}(\cdot) - \phi_m(\cdot) \right)
w_m \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{ \partial \epsilon_k }{ \partial \delta_1 }
= \; & - \phi_1(\cdot) \; w_1 \\
\frac{ \partial \epsilon_k }{ \partial \delta_m }
= \; & \phi_m(\cdot) \left( w_{m-1} - w_m \right)
\; \forall \; m \in \{2, \ldots, M-1 \} \setminus m^* \\
\frac{ \partial \epsilon_k }{ \partial \delta_M }
= \; & \phi_M(\cdot) \; w_{M-1} \\
\text{with } \phi_m
\equiv \; & \phi \left( \beta_0 +
\sum_{ j \in \{1, \ldots, K \} \setminus k } \beta_j x_j +
\delta_m \right)
\; \forall \; m = 1, \ldots, M
\label{eq:probElaIntPhiM}
\end{align}
and $x_0 \equiv 1$.
%=============================== semElaInt: Logit =============================
\subsubsection{Logistic regression}
The gradients are:
\begin{align}
\frac{ \partial \epsilon_k }{ \partial \beta_j }
= \; & x_j \sum_{m=1}^{M-1} \left(
\frac{ \exp_{m+1}(\cdot) }{ ( 1 + \exp_{m+1}(\cdot) )^2 }
- \frac{ \exp_{m}(\cdot) }{ ( 1 + \exp_{m}(\cdot) )^2 }
\right) w_m \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k \nonumber \\
\frac{ \partial \epsilon_k }{ \partial \delta_1 }
= \; & - \frac{ \exp_{1}(\cdot) }{ ( 1 + \exp_{1}(\cdot) )^2 } \; w_1 \\
\frac{ \partial \epsilon_k }{ \partial \delta_m }
= \; & \frac{ \exp_{m}(\cdot) }{ ( 1 + \exp_{m}(\cdot) )^2 }
\left( w_{m-1} - w_m \right)
\; \forall \; m \in \{2, \ldots, M-1 \} \setminus m^* \\
\frac{ \partial \epsilon_k }{ \partial \delta_M }
= \; & \frac{ \exp_{M}(\cdot) }{ ( 1 + \exp_{M}(\cdot) )^2 } \; w_{M-1}
\end{align}
with $x_0 \equiv 1$
and $\exp_m; m = 1, \ldots, M$ as defined in Table~\ref{tab:semiInt}.
%================================ semElaInt: MNL =============================
\subsubsection{Multinomial logistic regression}
The gradients are:
\begin{align}
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \beta_{j,o} }
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\frac{ \partial \epsilon_{k,\mathcal{P}} }{ \partial \delta_{m,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial \epsilon_{k,p}^* }{ \partial \delta_{m,o} }
\; \forall \; m \in \{ 1, \ldots, M \} \setminus m^*;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{\partial \epsilon_{k,p}^*}{\partial \beta_{j,o}}
= \; & x_j \sum_{m=1}^{M-1}
\Big( \pi_{p,m} \pi_{o,m} - \pi_{p,m+1} \pi_{o,m+1} \\
& \qquad \qquad - \Delta_{op} \left( \pi_{p,m} - \pi_{p,m+1} \right) \Big)
w_m \nonumber \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \nonumber \\
\frac{\partial \epsilon_{k,p}^*}{\partial \delta_{1,o}}
= \; & \left( \pi_{p,1} \pi_{o,1} - \Delta_{o,p} \; \pi_{p,1} \right) w_1 \\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \nonumber \\
\frac{\partial \epsilon_{k,p}^*}{\partial \delta_{m,o}}
= \; & \left( \pi_{p,m} \pi_{o,m} - \Delta_{o,p} \; \pi_{p,m} \right)
\left( w_m - w_{m-1} \right) \\
& \forall \; m \in \{ 2, \ldots, M-1 \} \setminus m^*; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \nonumber \\
\frac{\partial \epsilon_{k,p}^*}{\partial \delta_{M,o}}
= \; & - \left( \pi_{p,M} \pi_{o,M} - \Delta_{o,p} \; \pi_{p,M} \right) w_{M-1}\\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^*, \nonumber \\
\pi_{p,m}
\equiv \; & \frac{ \exp_{m,p} }{ \sum_{o=1}^P \exp_{m,o} }
\; \forall \; p = 1, \ldots, P; m = 1, \ldots, M,
\end{align}
$x_0 \equiv 1$, and $\exp_{m,p}; m = 1, \ldots, M; p = 1, \ldots, P$
as defined in Table~\ref{tab:semiInt}.
%+++++++++++++++++++++++++++++++++ effectInt +++++++++++++++++++++++++++++++++
\subsection{Gradients of effects of continuous covariates when they change between intervals}
\label{sec:gradEffectInt}
%================================ effectInt: LPM ==============================
\subsubsection{Linear probability model}
If the regression equation includes
only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial E_{k,lr}}{\partial \beta_j} &= 0
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial E_{k,lr}}{\partial \beta_k} &= \bar{x}_{kl} - \bar{x}_{kr}.
\end{align}
If the regression equation additionally includes
a quadratic term of the covariate of interest,
there is one additional gradient:
\begin{align}
\frac{\partial E_{k,lr}}{\partial \beta_{k+1}}
& = \overline{x^2_{kl}} - \overline{x^2_{kr}}.
\end{align}
%=========================== effectInt: Probit ===============================
\subsubsection{Probit regression}
If the regression equation includes
only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial E_{k, lr}}{\partial \beta_j}
& = \left( \phi_l - \phi_r \right) x_j
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial E_{k, lr}}{\partial \beta_k}
& = \phi_l \; \bar{x}_{kl} - \phi_r \; \bar{x}_{kr} \\
\text{with } \phi_n
& \equiv \phi\left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus k } \beta_j x_j
+ \beta_k \bar{x}_{kn} \right)
\; \forall \; n \in \{ l, r \}
\end{align}
and $x_0 \equiv 1$.
If the regression equation additionally includes
a quadratic term of the covariate of interest,
we have:
\begin{align}
\phi_n
\equiv \; & \phi\left( \beta_0
+ \sum_{ j \in \{ 1, \ldots, K \} \setminus \{ k, k + 1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kn} + \beta_{k+1} \overline{x^2}_{kn} \right) \\
& \forall \; n \in \{ l, r \} \nonumber
\end{align}
and there is one additional gradient:
\begin{align}
\frac{\partial E_{k, lr}}{\partial \beta_{k+1}}
& = \phi_l \; \overline{x^2}_{kl} - \phi_r \; \overline{x^2}_{kr} .
\end{align}
%=============================== effectInt: Logit =============================
\subsubsection{Logistic regression}
If the regression equation includes
only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{\partial E_{k, lr}}{\partial \beta_j}
& = \left(
\frac{ \exp_l( \cdot ) }{ ( 1 + \exp_l( \cdot ) )^2 }
- \frac{ \exp_r( \cdot ) }{ ( 1 + \exp_r( \cdot ) )^2 } \right) x_j
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{\partial E_{k, lr}}{\partial \beta_k}
& = \frac{ \exp_l( \cdot ) }{ ( 1 + \exp_l( \cdot ) )^2 } \; \bar{x}_{kl}
- \frac{ \exp_r( \cdot ) }{ ( 1 + \exp_r( \cdot ) )^2 } \; \bar{x}_{kr} \\
\text{with } \exp_n
& \equiv \exp\left( \beta_0
+ \sum_{j \in \{ 1, \ldots, K \} \setminus k } \beta_j x_j
+ \beta_k \bar{x}_{kn} \right)
\; \forall \; n \in \{ l, r \}
\end{align}
and $x_0 \equiv 1$.
If the regression equation additionally includes
a quadratic term of the covariate of interest,
we have:
\begin{align}
\exp_n
\equiv \; & \exp\left( \beta_0
+ \sum_{ j \in \{ 1, \ldots, K \} \setminus \{ k, k + 1 \} } \beta_j x_j
+ \beta_k \bar{x}_{kn} + \beta_{k+1} \overline{x^2}_{kn} \right) \\
& \forall \; n \in \{ l, r \} \nonumber
\end{align}
and there is one additional gradient:
\begin{align}
\frac{\partial E_{k, lr}}{\partial \beta_{k+1}}
& = \frac{ \exp_l( \cdot ) }{ ( 1 + \exp_l( \cdot ) )^2 }
\; \overline{x^2}_{kl}
- \frac{ \exp_r( \cdot ) }{ ( 1 + \exp_r( \cdot ) )^2 }
\; \overline{x^2}_{kr} .
\end{align}
%================================ effectInt: MNL ==============================
\subsubsection{Multinomial logistic regression}
If the regression equation includes
only a linear term of the covariate of interest,
the gradients are:
\begin{align}
\frac{ \partial E_{k,lr,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial E_{k,lr,p}^* }{ \partial \beta_{j,o} }
\; \forall \; j = 0, \ldots, K;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\frac{\partial E_{k, lr, p}^*}{\partial \beta_{j,o}}
= \; & \Big( \pi_{p,r} \; \pi_{o,r} - \pi_{p,l} \; \pi_{o,l}
- \Delta_{o,p} \left( \pi_{p,r} - \pi_{p,l} \right) \Big) x_j \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^* \nonumber \\
\frac{\partial E_{k, lr, p}^*}{\partial \beta_{k,o}}
= \; & \left( \pi_{p,r} \; \pi_{o,r} - \Delta_{o,p} \; \pi_{p,r} \right)
\bar{x}_{kr}
- \left( \pi_{p,l} \; \pi_{o,l} - \Delta_{o,p} \; \pi_{p,l} \right)
\bar{x}_{kl} \\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^* \nonumber \\
\text{with } \pi_{p,n}
\equiv & \frac{ \exp\left(
\beta_{0,p}
+ \sum_{ j \in \{1, \ldots, K \} \setminus k } \beta_{j,p} x_j
+ \beta_{k,p} \bar{x}_{k,n} \right) }{
\sum_{o=1}^P \exp\left( \beta_{0,o}
+ \sum_{ j \in \{1, \ldots, K \} \setminus k } \beta_{j,o} x_j
+ \beta_{k,o} \bar{x}_{k,n} \right)}\\
& \forall \; p = 1, \ldots, P; n \in \{ l, r \}, \nonumber
\end{align}
$x_0 \equiv 1$, and $\Delta_{o,p}$ denoting Kronecker's Delta
with $\Delta_{o,p} = 1 \; \forall \; o = p$
and $\Delta_{o,p} = 0 \; \forall \; o \neq p$.
If the regression equation additionally includes
a quadratic term of the covariate of interest,
we have:
\begin{align}
\pi_{p,n}
\equiv & \frac{ \exp\left(
\beta_{0,p}
+ \sum_{ j \in \{1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,p} x_j
+ \beta_{k,p} \bar{x}_{k,n} + \beta_{k+1,p} \overline{x^2}_{k,n} \right) }{
\sum_{o=1}^P \exp( \beta_{0,o}
+ \sum_{ j \in \{1, \ldots, K \} \setminus \{ k, k+1 \} } \beta_{j,o} x_j
+ \beta_{k,o} \bar{x}_{k,n} + \beta_{k+1,o} \overline{x^2}_{k,n} )}\\
& \forall \; p = 1, \ldots, P; n \in \{ l, r \} \nonumber
\end{align}
and there are $P \; ( P - 1 )$ additional gradients:
\begin{align}
\frac{\partial E_{k, lr, p}}{\partial \beta_{k+1,o}} = \;
& \left( \pi_{p,r} \; \pi_{o,r} - \Delta_{o,p} \; \pi_{p,r} \right)
\bar{x}_{kr}^2
- \left( \pi_{p,l} \; \pi_{o,l} - \Delta_{o,p} \; \pi_{p,l} \right)
\bar{x}_{kl}^2 \\
& \forall \; p = 1, \ldots, P;
o \in \{ 1, \ldots, P \} \setminus p^* . \nonumber
\end{align}
%+++++++++++++++++++++++++++++++++ effectCat +++++++++++++++++++++++++++++++++
\subsection{Gradients of grouped and re-based effects of categorical and interval-coded covariates}
\label{sec:gradEffectCat}
%================================ effectCat: LPM ==============================
\subsubsection{Linear probability model}
The gradients are:
\begin{align}
\frac{ \partial E_{k,lr} }{ \partial \beta_j }
= \; & 0 \; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{ \partial E_{k,lr} }{ \partial \delta_m }
= \; & D_{ml} - D_{mr}
\; \forall \; m \in \{1, \ldots, M \} \setminus m^*
\end{align}
%=========================== effectCat: Probit ===============================
\subsubsection{Probit regression}
The gradients are:
\begin{align}
\frac{ \partial E_{k,lr} }{ \partial \beta_j }
= \; & \left( \phi_l(\cdot) - \phi_r(\cdot) \right) x_j
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k \\
\frac{ \partial E_{k,lr} }{ \partial \delta_m }
= \; & \phi_l(\cdot) \; D_{ml} - \phi_r(\cdot) \; D_{mr}
\; \forall \; m \in \{1, \ldots, M \} \setminus m^* \\
\text{with } \phi_n(\cdot)
\equiv \; & \phi\left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{mn} \right) \\
& \forall \; n \in \{ r, l \} \nonumber
\end{align}
and $x_0 \equiv 1$.
%=============================== effectCat: Logit =============================
\subsubsection{Logistic regression}
The gradients are:
\begin{align}
\frac{ \partial E_{k,lr} }{ \partial \beta_j }
= \; & \left(
\frac{ \exp_l(\cdot) }{ \left( 1 + \exp_l(\cdot) \right)^2 }
- \frac{ \exp_r(\cdot) }{ \left( 1 + \exp_r(\cdot) \right)^2 }
\right) x_j \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k \nonumber \\
\frac{ \partial E_{k,lr} }{ \partial \delta_m }
= \; & \frac{ \exp_l(\cdot) }{ \left( 1 + \exp_l(\cdot) \right)^2 }
\; D_{ml}
- \frac{ \exp_r(\cdot) }{ \left( 1 + \exp_r(\cdot) \right)^2 }
\; D_{mr} \\
& \forall \; m \in \{1, \ldots, M \} \setminus m^* \nonumber \\
\text{with } \exp_n(\cdot)
\equiv \; & \exp\left( \beta_0
+ \sum_{j \in \{ 1, \ldots , K \} \setminus k} \beta_j x_j
+ \sum_{m=1}^M \delta_m D_{mn} \right) \\
& \forall \; n \in \{ r, l \} \nonumber
\end{align}
and $x_0 \equiv 1$.
%================================ effectCat: MNL ==============================
\subsubsection{Multinomial logistic regression}
The gradients are:
\begin{align}
\frac{ \partial E_{k,lr,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial E_{k,lr,p}^* }{ \partial \beta_{j,o} }
\; \forall \; j \in \{ 0, \ldots, K \} \setminus k;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\frac{ \partial E_{k,lr,\mathcal{P}} }{ \partial \beta_{j,o} }
= \; & \sum_{p \in \mathcal{P}}
\frac{ \partial E_{k,lr,p}^* }{ \partial \delta_{m,o} }
\; \forall \; m \in \{ 1, \ldots, M \} \setminus m^*;
o \in \{ 1, \ldots, P \} \setminus p^* \\
\text{with } \frac{\partial E_{k, lr, p}^*}{\partial \beta_{j,o}} = \; &
\left( \pi_{p,r} \; \pi_{o,r} - \pi_{p,l} \; \pi_{o,l}
- \Delta_{o,p} \left( \pi_{p,r} - \pi_{p,l} \right) \right) x_j \\
& \forall \; j \in \{ 0, \ldots, K \} \setminus k;
p = 1, \ldots, P, \nonumber \\
\frac{\partial E_{k, lr, p}^*}{\partial \delta_{m,o}} = \; &
\left( \pi_{p,r} \; \pi_{o,r} - \Delta_{o,p} \; \pi_{p,r} \right) D_{mr}
- \left( \pi_{p,l} \; \pi_{o,l} - \Delta_{o,p} \; \pi_{p,l} \right) D_{ml} \\
& \forall \; m \in \{1, \ldots, M \} \setminus m^*;
p = 1, \ldots, P, \nonumber \\
\pi_{p,n} \equiv &
\frac{
\exp\left( \beta_{0,p}
+ \sum_{j \in \{1, \ldots , K \} \setminus k } \beta_{j,p} x_j
+ \sum_{m=1}^M \delta_{m,p} D_{m,n} \right) }
{ \sum_{o=1}^P \exp\left( \beta_{0,o}
+ \sum_{j \in \{1, \ldots , K \} \setminus k } \beta_{j,o} x_j
+ \sum_{m=1}^M \delta_{m,o} D_{m,n} \right) }\\
& \forall \; p = 1, \ldots, P; n \in \{ l, r \}, \nonumber
\end{align}
and $x_0 \equiv 1$.
\clearpage
\section{Derivation of a binary probit model from an ordered probit model}
\label{sec:derivBinProbitOrderedProbit}
\begin{align}
\Pr( Y^* = 1 | X = x )
& = \Pr( Y \in \{ p^*, \ldots , P \} | X = x ) \\
& = \sum_{p = p^*}^{P} \Pr( Y = p | X = x ) \\
& = \sum_{p = p^*}^{P} \left(
\Phi\left( \mu_p - \sum_{j = 1}^K \beta_j x_j \right)
- \Phi\left( \mu_{p - 1} - \sum_{j = 1}^K \beta_j x_j \right) \right)\\
& = \sum_{p = p^*}^{P}
\Phi\left( \mu_p - \sum_{j = 1}^K \beta_j x_j \right)
- \sum_{p = p^*}^{P}
\Phi\left( \mu_{P - 1} - \sum_{j = 1}^K \beta_j x_j \right) \\
& = \sum_{p = p^*}^{P}
\Phi\left( \mu_p - \sum_{j = 1}^K \beta_j x_j \right)
- \sum_{p = p^* - 1}^{P-1}
\Phi\left( \mu_{p} - \sum_{j = 1}^K \beta_j x_j \right) \\
& = \Phi\left( \mu_P - \sum_{j = 1}^K \beta_j x_j \right)
- \Phi\left( \mu_{p^* - 1} - \sum_{j = 1}^K \beta_j x_j \right) \\
& = \Phi\left( \infty \right)
- \Phi\left( \mu_{p^* - 1} - \sum_{j = 1}^K \beta_j x_j \right) \\
& = 1 - \Phi\left( \mu_{p^* - 1} - \sum_{j = 1}^K \beta_j x_j \right) \\
& = \Phi\left( - \mu_{p^* - 1} + \sum_{j = 1}^K \beta_j x_j \right)
\end{align}
\clearpage
\section{Additional \R{} code}
\subsection{Loading and preparing data}
\label{sec:RCodeData}
Loading the data set:
<<eval=FALSE>>=
<<dataLoad>>
@
\noindent%
Creating a dummy variable for the presence of children in the household:
<<eval=FALSE>>=
<<dataKids>>
@
\noindent%
Creating dummy variables for interval-coding variable \code{age}:
<<eval=FALSE>>=
<<dataAgeInt>>
@
\noindent%
Creating an ordered categorical variable
that indicates three levels of labour force participation:
<<eval=FALSE>>=
<<dataLfp3>>
@
\subsection{Probit regressions with \code{age}
as linear covariate and with \code{age} as linear and quadratic covariate}
\label{sec:RCodeEstProbit}
Estimations and creating vectors with mean values of covariates:
<<eval=FALSE>>=
<<estProbit>>
@
\subsection{Logistic regressions with \code{age}
as interval-coded covariate}
\label{sec:RCodeEstLogitInt}
Estimation and creating a vector with mean values of covariates:
<<eval=FALSE>>=
<<estLogitInt>>
@
\subsection{Ordered probit regression with \code{age}
as linear and quadratic covariate}
\label{sec:RCodeEstOProbit}
Estimation:
<<eval=FALSE>>=
<<estOProbit>>
@
\subsection{Multinomial logistic regressions with \code{age}
as interval-coded covariate}
\label{sec:RCodeEstMLogitInt}
Estimation:
<<eval=FALSE>>=
<<estMLogitInt>>
@
\end{document}