A researcher in geriatrics designed a prospective study to investigate the effects of two interventions on the frequency of falls. One hundred subjects were randomly assigned to one of the two interventions: education only X1 = 0) and education plus aerobic exercise (X1 = 1). Subject were at least 65 years of age and in reasonably good health. Three variables considered to be important as control variables were (X2 : 0 = female; 1 = male), a balance index (X3), and a strength index (X4). The higher the balance index, the more stable is the subject; and the higher the strength index, the stronger the subject. Each subject kept a diary recording the number of falls (Y ) during the six months of study.

please use that information, figure out which poisson model better. why is better?

Call: glm(formula = Y ~ X1 + X2 + X3 + X4, family = poisson(), data = geriatric) Deviance Residuals: Min 10 Median 3Q Max -2. 1854 -0. 7819 -0. 2564 0. 5449 2. 3626 Coefficients : Estimate Std. Error z value Pr(> |z|) (Intercept) 0. 489467 0. 336869 1. 453 0. 14623 X1 -1. 069403 0. 133154 -8. 031 9. 64e-16 * * * X2 -0. 046606 0. 119970 -0. 388 0. 69766 X3 0. 009470 0. 002953 3.207 0. 00134 * * X4 0 . 008566 0 . 004312 1.986 0 . 04698 * Signif. codes: 0 '** *' 0. 001 '* * ' 0.01 '*' 0. 05 '.' 0.1 ' ' 1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 199.19 on 99 degrees of freedom Residual deviance: 108.79 on 95 degrees of freedom AIC: 377.29 Number of Fisher Scoring iterations: 5 (Intercept) X1 X2 X3 X4 0. 489467165 -1. 069402551 -0. 046606063 0. 009469987 0. 008565829 Residual deviance = 108.7899 on 95 df Dispersion parameter = 1Call: glm(formula = Y ~ X1 + X3 + X4, family = poisson(), data = geriatric) Deviance Residuals: Min 1Q Median 3Q Max -2. 2152 -0.7512 -0. 2594 0. 5830 2. 2893 Coefficients : Estimate Std. Error z value Pr(>|z|) (Intercept) 0. 443890 0. 317289 1. 399 0. 16181 X1 1. 077770 0. 131415 -8. 201 2. 38e-16 * * * X3 0. 009471 0. 002957 3. 203 0. 00136 * * X4 0. 008979 0 . 004190 2. 143 0. 03209 Signif. codes: 0 '* * *' 0. 001 '* * ' 0. 01 ' * ' 0. 05 '.' 0.1 "'1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 199.19 on 99 degrees of freedom Residual deviance: 108.94 on 96 degrees of freedom AIC: 375 .44 Number of Fisher Scoring iterations: 5 (Intercept) X1 X3 X4 0. 44389020 -1. 07777033 0. 00947114 0. 00897921 Residual deviance = 108.9409 on 96 df Dispersion parameter = 1