Variable Combinations
Data
## weight age.years lat site
## 1 12615.13 41 2.690077 Site 1
## 2 12539.55 50 2.974857 Site 2
## 3 13753.97 21 -4.868759 Site 1
## 4 17269.26 50 -28.437431 Site 2
## 5 16945.27 16 -26.414605 Site 1
## 6 14723.24 39 -11.392111 Site 2Questions
1. What model syntax allows for the effect of age.years to be different at each level of site?
first = glm(weight~age.years+site+age.years:site)
second = glm(weight~age.years+site)
third = glm(weight~age.years*site)
fourth = glm(weight~poly(age.years,2)+siteThe First and Third (same model/design matrix, but different code syntax) allow the effect of slope to be different at each level of site. These are interactions b/w variables, such that the variables co-depend on each other.
The Second allows there to be a site-level differences but the same slope
The fourth also allows a site level difference of a quadratic effect, but the same quadratic effect
Main take away - all allow for different effects of age.years by site, but the First and Third let the effect/slope be different, while the other models just move the same slope up and down due to site.
2. Draw an x-y plot showing the relationship b/w weight and the additive effect of site and age.years. Assume a negative slope of age.years with weight and two levels of the variable site. Assume the intercept mle is 13000 and the estimated effect of siteSite2 = -2000. Label axes and slopes for each site.
glm(weight~age.years+site)The plot should have weight on the y-axis and age.years on the x-axis. There should be two lines, representing the slopes of age.years for each site. The slopes should be parallel with site 2 lower than site 1. The slope for site 2 should be lower on the y-axis than site 1.
3. Looking at the data table above, write out the design matrix for the two models below.
glm(weight~age.years+site)
glm(weight~age.years*site)## (Intercept) age.years siteSite 2
## 1 1 41 0
## 2 1 50 1
## 3 1 21 0
## 4 1 50 1
## 5 1 16 0
## 6 1 39 1## (Intercept) age.years siteSite 2 age.years:siteSite 2
## 1 1 41 0 0
## 2 1 50 1 50
## 3 1 21 0 0
## 4 1 50 1 50
## 5 1 16 0 0
## 6 1 39 1 394. When might it be good to assume an additive effect b/w a categorical and continuous variable over an interaction?
When there are many levels of the categorical variable and not a lot of replicate data at each or some levels. Also, when the hypothesis is an effect of difference by each level, but the same slope.
5. Define what each coefficient means. Make sure to make clear the units.
dat$age.yeras.sc = scale(dat$age.years,center=TRUE, scale=FALSE)
summary(glm(weight~age.yeras.sc*site,data=dat))##
## Call:
## glm(formula = weight ~ age.yeras.sc * site, data = dat)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14574.77 241.92 60.245 < 2e-16 ***
## age.yeras.sc -65.32 24.35 -2.682 0.00861 **
## siteSite 2 -188.06 342.00 -0.550 0.58367
## age.yeras.sc:siteSite 2 108.48 32.14 3.375 0.00107 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2908337)
##
## Null deviance: 313494594 on 99 degrees of freedom
## Residual deviance: 279200364 on 96 degrees of freedom
## AIC: 1778
##
## Number of Fisher Scoring iterations: 2Intercept = the mean weight of elephants at the observed mean age of elephants at site 1
age.yeras.sc = the one year change in mean weight of elephants at site 1
siteSite 2 = the mean weight effect difference of site 2 from site 1 (intercept) at the mean observed age of elephants.
age.yeras.sc:siteSite 2 - the change in the slope or one year change in mean weight of elephants at site 2 from site 1