Math 141. Lecture 24: Model Comparisons and The F-test. Albyn Jones 1. 1 Library jones/courses/141
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1 Math 141 Lecture 24: Model Comparisons and The F-test Albyn Jones 1 1 Library 304 jones@reed.edu jones/courses/141
2 Nested Models Two linear models are Nested if one (the restricted model) is obtained from the other (the full model) by setting some parameters to zero (i.e. removing terms from the model), or some other constraint on the parameters. We can compare nested models fit to the same dataset with the F test.
3 Example # Full Model Mfull <- lm(y X + W + Z + T, data = MyDataSet) # Restricted Model Mres <- lm(y X + W, data = MyDataSet ) Fitting the restricted model is equivalent to forcing β Z = β T = 0 in the full model.
4 Comparing Nested Models The crucial question is whether the residual sum of squares for the restricted model (RSS R ) is substantially larger than the residual sum of squares for the full model (RSS F ). R. A. Fisher worked out the distribution of a ratio of the two under the null hypothesis that the restricted model is correct, which typically corresponds to the statement that some parameters are zero. As usual, this story depends on the residuals having a at least an approximately normal distribution.
5 The F-Test Assuming model validity, the F-ratio (F is for Fisher, by the way) F dfn,df F = (RSS R RSS F )/(df R df F ) RSS F /df F has an F distribution with degrees of freedom (df N, df F ) if the restricted model is correct. Note: df N = df R df F, and df F and df R are residual df from the two models. Reject: if F > qf (.95, df N, df F ) Note: df R df F is always the number of constraints on the parameters that converts the full model to the restricted model.
6 The F density F(5,20) density density Rejection Region X
7 Analogy F(5,20) density vs Chisq(5)/5 density X F k,n is to χ 2 k as t n is to N(0, 1). The denominator estimates σ 2. If we knew σ 2, the ratio would have a χ 2 distribution.
8 Connection to the t Distribution: F 1,k is t 2 k lm(formula = ht18 ht2, data = Berkeley) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ht e Residual standard error: on 56 degrees of freedom F-statistic: on 1 and 56 DF, p-value: 2.2e-06 > 5.278ˆ2 [1]
9 Example, CPS wage data summary Call: lm(formula = wage race*sex + educ + age + union, data = CPS) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <.001 racew sexm educ <.001 age <.001 unionunion racew:sexm
10 Plot Residuals! fitted(cps.lmf) residuals(cps.lmf) CPS wage residuals What Next?
11 Example, CPS log(wage) data summary Call: lm(formula = log(wage) race*sex + educ + age + union, data = CPS) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) racew sexm educ <.01 age <.01 unionunion <.01 racew:sexm Residual standard error: on 527 df
12 Plot Residuals Again! fitted(cps.lmf) residuals(cps.lmf) CPS log wage residuals Better?
13 Looking for a parsimonius model? None of the coefficients for race, sex, and the race*sex interaction were statisticially significantly different from zero. Let s fit a restricted model, dropping those non-significant explanatory variables.
14 Example, CPS log(wage) Restricted Model Call: lm(formula = log(wage) educ + age + union, data = CPS) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <.01 educ <.01 age <.01 unionunion <.01 Residual standard error: on 530 df
15 Model Comparison > anova(cps.loglmr,cps.loglmf) Analysis of Variance Table Model 1: log(wage) educ + age + union Model 2: log(wage) race*sex + educ + age + union Res.Df RSS Df Sum of Sq F Pr(>F) e-08 Say WHAT? None of the omitted coefficients were statistically significantly different from 0! How can this happen?
16 The Null and Alternative Hypotheses What is H 0? The restricted model is correct. Informally: the restricted model fits as well as the full model. Formally: H 0 : coefficients for the omitted terms are all 0. Formally: H 1 : at least one omitted coefficient is not zero.
17 Important! Individual t-tests are testing a null hypothesis for a single coefficient H 0 : β = 0 given we have controlled for the other variables in the model!
18 What was missing? formula = log(wage) sex + educ + age + union, data = CPS) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) sexm <.001 educ <.001 age <.001 unionunion <.001
19 What happened? The race*sex interaction was a distraction! > cor(sex=="m",sex=="m" & race=="w") [1] Strongly correlated explanatory variables can be distractors, each does part of the work of predicting the response, neither seems important when the other is included.
20 Interpretation The coefficient for the dummy variable for Males was about.23. What does that mean? All other factors held equal, the difference between log(wage) for males and log(wage) for females is.23: Therefore log(w ) = OtherStuff +.23 sexm W = e OtherStuff +.23 sexm = e OtherStuff e.23 sexm The dummy variable sexm is 1 for males and 0 for females, so the difference is the multiplicative factor e Conclusion: Males with the same education level, age and Union status get paid about 26% more than corresponding females with the same covariate values.
21 R will try to prevent silliness > anova(cps.loglmr,cps.lmf) Analysis of Variance Table Response: log(wage) Df Sum Sq Mean Sq F value Pr(>F) educ <.001 age <.001 union <.001 Residuals Warning message: In anova.lmlist(object,...) : models with response "wage" removed because response differs from model 1
22 Michelson s Data, full model > summary(mf) Call: lm(formula = Speed Run, data = Michelson) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) Run Run Run Run
23 Michelson s Data, restricted model > Run1 <- Michelson$Run == 1 > summary(mr) Call: lm(formula = Speed Run1, data = Michelson) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 2.998e e < 2e-16 Run1TRUE 7.075e e
24 Model Comparison! > anova(mr,mf) Analysis of Variance Table Model 1: Speed Run1 Model 2: Speed Run Res.Df RSS Df Sum of Sq F Pr(>F) What was H 0, and what do we conclude?
25 Summary The F test compares nested models fit to the same dataset. It allows us to test hypotheses involving multiple parameters simultaneously. If you wish to conclude that a collection of coefficients are all zero, or none of a subset of your explanatory variables predict the response, an F-test is the appropriate tool.
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