III. Model Building. Model building: writing a model that will provide a good fit to a set of data & that will give good estimates of the mean value of y and good predictions of y for given values of the explanatory variables.
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“A social science theory is a reasoned and precise speculation about the answer to a research question, including a statement about why the proposed answer is correct.” • “Theories usually imply several more specific descriptive or causal hypotheses” (King et al., page 19).
A model is “a simplification of, and approximation to, some aspect of the world.” • “Models are never literally ‘true’ or ‘false,’ although good models abstract only the ‘right’ features of the reality they represent” (King et al., page 49).
Three approaches to model building • Begin with a linear model for its simplicity & as a rough approximation of the y/x relationships. • Begin with a curvilinear model to capture the complexities of the y/x relationships. • Begin with a model that incorporates linearity &/or curvilinearity in y/x relationships according to theory & observation.
The predominate approach used to be to start with a simple model & to test it against progressively more complex models. • This approach suffers, however, from the problem associated with omitted variables in the simpler models. • Increasingly common, then, is the approach of starting with more complex models & testing against simpler models (Greene, Econometric Analysis, pages 151-52).
The point of departure for model-building is trying to grasp how the outcome variable y varies as the levels of an explanatory variable change. • We have to know how to write a mathematical equation to model this relationship.
In what follows, let’s pretend that we’ve already done careful univariate & bivariate exploratory data analysis via graphs & numerical summaries (although, on the other hand, the exercise requires here & there that we didn’t do such careful groundwork…).
Suppose we want to model a person’s performance on an exam, y, as a function of a single explanatory variable, x, the person’s amount of study time. • It may be that the person’s score, y, increases in a straight line as the amount of study time increases from 1 to 6 hours.
What, though, if the range of sample hours increased to 10 hours or more: would a straight-line model continue to be satisfactory? • Quite possibly, the increase in exam score for a unit increase in study time would decrease, causing some amount of curvature in the y/x relationship.
: value of y when x’s equal 0; shifts the quadratic parabola up or down the y-intercept : the slop of y on x when x=0 (which we don’t really care about & won’t interpret) : negative, degree of downward parabola; positive, degree of upward parabola
Let’s continue with this model-building strategy but change the substantive topic. • We’ll focus on the relationship of average hourly wage to a series of explanatory variables (e.g., education & job tenure with the same employer). • Let’s explore the relationship.
Median Band Regression & Lowess Smoothing • Median band regression (scatter mband y x) & lowess smoothing (lowess y x) are two very helpful tools for detecting (1) how a model fits or doesn’t fit to particular segments of the x –values (e.g., poorer to richer persons) & (2) thus non-linearity. • Hence they’re really useful at all stages of exploratory data analysis. • Another option: locpoly y x1
What did the graphs say about the relationship of wage to years of education? • Let’s answer this question more precisely by predicting the direction & magnitude of the wage/education relationship at specific levels of education, identified via ‘su x, detail’ and/or our knowledge of the issue:
Remember, moreover, that Adj R2 was just .198 for this model. • Obviously there are other relevant explanatory variables. Not only do we need to identify them, but we also need to ask: are they independent & linear? independent & curvilinear? or are they interactional?
Interaction Effects • Interaction: the effect of a 1-unit change in one explanatory variable depends on the level of another explanatory variable. • With interaction, both the y-intercept & the regression slope change; i.e. the regression lines are not parallel.
Let’s interpret the model. • If the interaction term tests significant, we don’t interpret its base variables? • Why not? Each base variable represents its y/x slope when the other x=0. We don’t care about this.
To interpret the interaction term, we use ‘su x1, d’ & our knowledge of the subject to identify key levels of educ & tenure (or use one SD above mean, mean, & one SD below mean): • Then we predict the slope-effect of educXtenure on wage at the specified levels, as follows:
How the interaction of mean education with varying levels of tenure relates to average hourly wage: • . lincom educ + (educXtenure*2) • wage Coef. Std. Err. t P>t [95% Conf. Interval] • (1) -.0372278 .062391 -0.60 0.551 -.1597961 .0853406 • . lincom educ + (educXtenure*10) • wage Coef. Std. Err. t P>t [95% Conf. Interval ] • (1) .142818 .0221835 6.44 0.000 .0992382 .1863978 • . lincom educ + (educXtenure*18) • wage Coef. Std. Err. t P>t [95% Conf. Interval] • (1) .3678751 .0503567 7.31 0.000 .2689484 .4668019
How the interaction of mean tenure with varying levels of education relates to average hourly wage: • . lincom tenure + (8*educXtenure) • wage Coef. Std. Err. t P>t [95% Conf. Interval] • (1) .0978065 0303746 3.22 0.001 .0381351 .157478 • . lincom tenure + (12*educXtenure) • wage Coef. Std. Err. t P>t [95% Conf. Interval] • (1) .1878294 .0184755 10.17 0.000 .1515338 .224125 • . lincom tenure + (20*educXtenure) • wage Coef. Std. Err. t P>t [95% Conf. Interval] • (1) .3678751 .0503567 7.31 0.000 .2689484 .4668019
With significant interaction, to repeat, both the regression coefficient & the y-intercept change as the levels of the second interacting variable change. • That is, the regression slopes are unequal. What does this mean in the model for average hourly wage?
Let’s assess the model’s fit. • Let’s conduct a test of nested models, comparing this new, ‘full’ model to each of the previous, ‘reduced’ models.
Did adding educXtenure, educ2 & tenure2 boost the model’s variance-explaining power by a statistically significant margin? • . test educXtenure educ2 tenure2 • ( 1) educXtenure = 0 • ( 2) educ2 = 0 • ( 3) tenure2 = 0 • F( 3, 520) = 17.47 • Prob > F = 0.0000
Did adding educ2 & tenure2 boost the model’s variance-explaining power by a statistically significant margin over the interaction model? . test educ2 tenure2 ( 1) educ2 = 0 ( 2) tenure2 = 0 F( 2, 520) = 18.48 Prob > F = 0.0000
Valid testing of nested models To conduct a valid test of nested models: • the number of observations for both the complete & reduced models must be equal; • the functional form of y must be the same (e.g., we can’t compare outcome variable ‘wage’ to outcome variable ‘log-wage’).
Comparing non-nested models • How do we compare non-nested models (i.e. models with the same number of explanatory variables), or nested models that don’t meet the criteria for comparative testing? • Use either the AICor BIC test statistics: the smaller the score, the better the model fits. • Download the ‘fitstat’ command (see Long/Freese, Regression Models for Categorical Dependent Variables).
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