1. Logistic Regression with “Grouped” Data
Lobster Survival by Size in a Tethering Experiment
Source: E.B. Wilkinson, J.H. Grabowski, G.D. Sherwood, P.O. Yund (2015). "Influence of Predator Identity on the Strength of Predator Avoidance Response in Lobsters," Journal of Experimental Biology and Ecology, Vol. 465, pp

2. Data Description
Experiment involved 159 juvenile lobsters in a Saco Bay, Maine
Outcome: Whether or not lobster survived predator attack in Tethering Experiment
Predictor Variable: Length of Carapace (mm).
Lobsters grouped in m = 11 groups of width of 3mm (27 to 57 by 3)

3. Models Data: (Yi,ni) i=1,…,m
Distribution: Binomial at each Size Level (Xi)
Link Function: Logit: log(pi/(1-pi)) is a linear function of Predictor (Size Level)
3 Possible Linear Predictors:
log(pi/(1-pi)) = a (Mean is same for all Sizes, No association)
log(pi/(1-pi)) = a + bXi (Linearly related to size)
log(pi/(1-pi)) = b1Z1 +…+ bm-1Zm-1 + bmZm Zi = 1 if Size Level i, 0 o.w. This allows for m distinct logits, without a linear trend in size (aka Saturated model)

4. Probability Distribution & Likelihood Function - I

5. Maximum Likelihood Estimation – Model 1

6. Maximum Likelihood Estimation – Model 3

7. Model 2 – R Output
glm(formula = lob.y ~ size, family = binomial("logit"))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) e-08 ***
size e-09 ***

8. Evaluating the log-likelihood for Different Models

9. Deviance and Likelihood Ratio Tests
-2*(log Likelihood of Model – log Likelihood of Saturated Model)
Degrees of Freedom = # of Parameters in Saturated Model - # in model
When comparing a Complete and Reduced Model, take difference between the Deviances (Reduced – Full)
Under the null hypothesis, the statistic will be chi-square with degrees of freedom = difference in degrees of freedoms for the 2 Deviances (number of restrictions under null hypothesis)
Deviance can be used to test goodness-of-fit of model.

10. Deviance and Likelihood Ratio Tests

11. Pearson Chi-Square Test for Goodness-of-Fit

12. Pearson Chi-Square Test for Goodness-of-Fit
Even though some of the group sample sizes are small, and some Expected cell
Counts are below 5, it is clear that Model 2 provides a Good Fit to the data

13. Residuals

14. Residuals

15. Computational Approach for ML Estimator

16. Estimated Variance-Covariance for ML Estimator

17. ML Estimate, Variance, Standard Errors
> mod2 <- glm(lob.y ~ size, family=binomial("logit"))
> summary(mod2)
Call: glm(formula = lob.y ~ size, family = binomial("logit"))
Deviance Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) e-08 ***
size e-09 ***
Null deviance: on 10 degrees of freedom
Residual deviance: on 9 degrees of freedom
AIC: 32.24
logLik(mod2)
'log Lik.' (df=2)