## 8.8 Likelihood ratio test vs. Wald test

As with previous chapters, Wald tests for p-values were used in this chapter. However, likelihood ratio (LR) tests are, in general, more powerful. True LR tests are not possible with svyglm() objects since they were not fit using maximum likelihood . However, the function regTermTest() can be used to carry out a “working” LR test for weighted linear, logistic, or Cox regression models (the Rao-Scott LR test) to compare any two nested models, similar to anova() which we used in previous chapters.

regTermTest() can therefore obtain an overall Type 3 Wald or working LR test for a categorical predictor with more than two levels. To get a test for a single level of a categorical predictor, first create indicator variables for the levels of that predictor as described in Section 6.18.

Example 8.1 (continued): Use a working LR test to test the overall significance of race/ethnicity in the weighted adjusted linear regression model for fasting glucose. For comparison, also compute the Wald test.

# Model fit previously
fit.ex8.1 <- svyglm(LBDGLUSI ~ BMXWAIST + smoker + RIDAGEYR +
RIAGENDR + race_eth + income,
family=gaussian(), design=design.FST.nomiss)
# Working LR test for race_eth
regTermTest(fit.ex8.1,
test.terms = ~ race_eth,
df = degf(fit.ex8.1$survey.design), method = "LRT") ## Working (Rao-Scott+F) LRT for race_eth ## in svyglm(formula = LBDGLUSI ~ BMXWAIST + smoker + RIDAGEYR + RIAGENDR + ## race_eth + income, design = design.FST.nomiss, family = gaussian()) ## Working 2logLR = 11.29 p= 0.042 ## (scale factors: 1.8 0.93 0.31 ); denominator df= 15 # Wald test for race_eth regTermTest(fit.ex8.1, test.terms = ~ race_eth, df = degf(fit.ex8.1$survey.design),
method = "Wald")
## Wald test for race_eth
##  in svyglm(formula = LBDGLUSI ~ BMXWAIST + smoker + RIDAGEYR + RIAGENDR +
##     race_eth + income, design = design.FST.nomiss, family = gaussian())
## F =  3.003  on  3  and  15  df: p= 0.064

Conclusion: Based on the likelihood-ratio test, race/ethnicity is significantly associated with fasting glucose, after adjusting for the other variables in the model (p = .042). As previously mentioned, LRTs are generally more powerful than Wald tests, which means lower p-values. In this example, that is the case, with the Wald test p-value being .064.

### References

———. 2010. Complex Surveys: A Guide to Analysis Using r: A Guide to Analysis Using r. John Wiley; Sons.
Lumley, Thomas, and Alastair Scott. 2013. “Partial Likelihood Ratio Tests for the Cox Model Under Complex Sampling.” Statistics in Medicine 32 (1): 110–23. https://doi.org/10.1002/sim.5492.
———. 2014. “Tests for Regression Models Fitted to Survey Data.” Australian & New Zealand Journal of Statistics 56 (1): 1–14. https://doi.org/10.1111/anzs.12065.
Rao, J. N. K., and A. J. Scott. 1984. “On Chi-Squared Tests for Multiway Contingency Tables with Cell Proportions Estimated from Survey Data.” The Annals of Statistics 12 (1): 46–60. https://www.jstor.org/stable/2241033.