Comparing coefficients of nested nonlinear probability models using khb Ulrich Kohler Joined Work with Kristian B. Karlson and Anders Holm 9 th German Stata Users Group Meeting Bamberg 01 July 2011 U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 1 / 25
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Comparing coefficients of nested nonlinearprobability models using khb
Ulrich Kohler
Joined Work with Kristian B. Karlson and Anders Holm
9th German Stata Users Group MeetingBamberg
01 July 2011
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 1 / 25
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 2 / 25
Introduction
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 3 / 25
Introduction
Reasons to compare
Depvar (Y)Keyvar (X)bX |Z
Mediator (Z)
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 4 / 25
Introduction
Reasons to compare
Depvar (Y)Keyvar (X)bX
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 4 / 25
Introduction
Reasons to compare
Depvar (Y)Keyvar (X)bX
bX |Z
Mediator (Z)
bX − bX |Z
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 4 / 25
Introduction
Reasons to compare
Depvar (Y)
Keyvar (X)
bX
bX |Z
Control (Z)
bX − bX |Z
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 4 / 25
Introduction
Reasons to compare
Depvar (Y)Keyvar (X)bX
bX |Z
Control (Z)
bX − bX |Z
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 4 / 25
Introduction
The problem
We are interested in obtaining βR − βF from the following models forlatent Y ∗:
Y ∗ = αF + βF X + γF Z + δF C + ε (1)Y ∗ = αR + βRX + δRC + ε (2)
Having ovserved Y with value 0 if Y ∗ < τ and 1 if Y ∗ ≥ τ we canobtain the logit/probit estimates with
bF =βF
σFand bR =
βR
σR(3)
Note: We identify the underlying coefficients of interest relative to ascale unknown to us.
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 5 / 25
The KHB-method
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 6 / 25
The KHB-method
General idea
The KHB-method extracts from Z the information that is not containedin X . This is done by calculating the residuals of a linear regression ofZ on X , i.e,
R = Z − (a + bX ) , (4)
where a and b are the estimated regression parameters of a linearregression.
Instead of using equation (2) we then use
Y ∗ = α̃R + β̃RX + γ̃RR + δ̃RC + ε . (5)
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 7 / 25
The KHB-method
Difference of coefficients
As R and Z differ only in the component in Z that is correlated with X ,model (1) is no more predictive than model (5), and consequently theresiduals have the same standard deviation so that
σ̃R = σF (6)
As β̃R = βR we can write
b̃R − bF =β̃R
σ̃R− βF
σF=βR − βF
σF. (7)
Hence, the difference obtained reflects the difference searched dividedby some common scale.
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 8 / 25
The KHB-method
Derived statistics
Confounding ratiob̃R
bF=
βRσFβFσF
=βR
βF, (8)
Counfounding percentage
100 · b̃R − bF
b̃R= 100 ·
βRσF− βF
σFβRσF
= 100 · βR − βF
βR, (9)
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 9 / 25
The KHB-method
Significance test for the difference in effects
Analyitcally derived standard errors for the difference in effectsexist.Based on the delta method (Sobel, 1982).Simple for one X and ond Z but fairly complicated for situationswith more than one X , Z .Karlson et al. (2010) has more details; also see our Stata Journalpublications (in Press)
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 10 / 25
The command khb
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 11 / 25
The command khb
Syntax
khb model-type depvar key-vars ‖ mediator-vars[
if][
in][, options
]model-type can be any of regress, logit, ologit, probit,oprobit, cloglog, slogit, scobit, rologit, clogit, andmlogit.
key-vars may contain factor variables
aweights, fweights, iweights, and pweights are allowed if theyare allowed for the specified model type.
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 12 / 25
The command khb
Options (most important ones)
options descriptionconcomitant(varlist) concomitantsdisentangle disentangle difference of effectssummary summary of decompositionvce(vcetype) robust or cluster clustvarape decomposition using avg. partial effectsverbose show restricted and full modelkeep keep residuals of mediators
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 13 / 25
Application
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 14 / 25
Application
Preliminaries
Examples from educational sociologySubset of Danish National Longitudinal Survey (DLSY).Reproduce analysis presented by Karlson and Holm (2011).
. use dlsy_khb, clear
. describe
Contains data from dlsy_khb.dtaobs: 1,896vars: 8 17 Jan 2011 10:26size: 49,296 (99.9% of memory free)
storage display valuevariable name type format label variable label
edu byte %20.0g edu Educational attainmentupsec byte %10.0g yesno Complete upper secondary
education (Gymnasium)univ byte %13.0g yesno Complete University educationfgroup byte %9.0g fgroup Father´s social group/classfses float %9.0g Father´s SES, standardized with
mean 0 and sd 1abil double %10.0g Standardized ability measure,
with mean 0 and sd 1intact byte %9.0g yesno Intact familyboy byte %9.0g yesno Boy
Sorted by:
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 15 / 25
Application
Basic use
. khb logit univ fses || abil, c(intact boy)
Decomposition using the KHB-Method
Model-Type: logit Number of obs = 1896Variables of Interest: fses Pseudo R2 = 0.19Z-variable(s): abilConcomitant: intact boy
N 1896 1896Conf.-Ratio 1.349 1.411Conf.-Perc. 25.88 29.15
t statistics in parentheses* p<0.05, ** p<0.01, *** p<0.001
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 23 / 25
References
Outline
1 Introduction
2 The KHB-method
3 The command khb
4 Application
5 References
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 24 / 25
References
References
Karlson, K. B. and A. Holm. 2011. Decomposing primary andsecondary effects: A new decomposition method. Research inStratification and Social Mobility 29: XXXX.
Karlson, K. B., A. Holm, and R. Breen. 2010. Comparing regressioncoefficients between models using logit and probit: a new method.Unpublished paper (currently under review).
Sobel, M. E. 1982. Asymptotic confidence intervals for indirect effectsin structural equation models. In Sociological Methodology 1982, ed.L. S., 290–312. Washington D.C.: American SociologicalAssociation.
U. Kohler (WZB) Comparing Coeficients with khb 01 July 2011 25 / 25