Population Profiles and Response profiles

The responses we're modeling are four cell counts:

Maximum Likelihood Analysis of Variance

As with ANOVA, we are breaking down the variability across factors. More specifically, the main effect TREATMENT tests if the subjects are evenly distributed over the two levels of this variable. The null hypothesis assumes that they are, and the alternative assumes that they are not. Based on the output below, are the subjects evenly distributed across the placebo and the vitamin C conditions?

The main effect RESPONSE tests if the subjects are evenly distributed over the two levels of this variable. The Wald statistic \(X^2=98.12\) with df = 1 indicates highly uneven distribution.

The LIKELIHOOD RATIO (or the deviance statistic) tells us about the overall fit of the model. Does this value look familiar? Does the model of independence fit well?

Analysis of Maximum Likelihood Estimates

Gives the estimates of the parameters. We focus on the estimation of odds.

For example,

\(\lambda_1^A=\lambda_{placebo}^{TREATMENT}=0.00358=-\lambda_{ascobic}^{TREATMENT}=-\lambda_2^A\)

What are the odds of getting cold? Recall that PROC CATMOD, by default, uses zero-sum constraints. Based on this output the log odds are

\begin{align}
\text{log}(\mu_{i1}/\mu_{i2}) &=\text{log}(\mu_{i1})-\text{log}(\mu_{i2})\\
&=[\lambda+\lambda_{placebo}^{TREATMENT}+\lambda_{cold}^{RESPONSE}]-[\lambda+\lambda_{placebo}^{TREATMENT}+\lambda_{nocold}^{RESPONSE}]\\
&=\lambda_{cold}^{RESPONSE}-\lambda_{nocold}^{RESPONSE}\\
&=\lambda_{cold}^{RESPONSE}+\lambda_{cold}^{RESPONSE}\\
&=-0.7856-0.7856=-1.5712\\
\end{align}

So the odds are \(\exp(-1.5712) = 0.208\). Notice that the odds are the same for all rows.

Model Information

Gives assumptions about the model. In our case, tells us that we are modeling the log, i.e., the LINK FUNCTION, of the responses, i.e. DEPENDENT VARIABLE, that is counts which have a Poisson distribution.

 

Class Level Information

Information on the categorical variables in our model

Criteria For Assessing Goodness of Fit

Gives the information on the convergence of the algorithm for MLE for the parameters, and how well the model fits. What is the value of the deviance statistic \(G^2\)? What about the Pearson \(X^2\)? Note that they are the same as what we got when we did the chi-square test of independence; we will discuss "scaled" options later with the concept of overdispersion. Note also the value of the log-likelihood for the fitted model, which in this case is independence.

Analysis of Parameter Estimates

Gives individual parameter estimates. Recall that PROC GENMOD does not use zero-sum constraints but fixes typically the last level of each variable to be equal to zero:

\(\hat{\lambda}=4.75,\ \hat{\lambda}^A_1=0.0072,\ \hat{\lambda}^A_2=0,\ \hat{\lambda}^B_1=-1.5712,\ \hat{\lambda}^B_2=0\)

Note:The scale parameter was held fixed.

So, we can compute the estimated cell counts for having cold given vitamin C based on this model:

\(\mu_{21}=\exp(\lambda+\lambda_2^A+\lambda_1^B)=\exp(4.745+0-1.5712)=23.91 \)

 

What about the other cells: \(\mu_{11}\), \(\mu_{12}\), and \(\mu_{22}\)?

What are the estimated odds of getting a cold?

\(\exp(\lambda_1^B-\lambda_2^B)=\exp(-1.571-0)=0.208\)

What about the odds ratio?

LR Statistics for Type 3 Analysis

Testing the main effects is similar to the Maximum Likelihood Analysis of Variance in the CATMOD, except this is the likelihood ratio (LR) statistic and not the Wald statistic. When the sample size is moderate or small, the likelihood ratio statistic is the better choice. From the CATMOD output, the chi-square Wald statistic was 98.12, with df=1, and we reject the null hypothesis. Here, we reach the same conclusion, except, that LR statistic is 130.59 with df=1.

Observation Statistics

Gives information on expected cell counts and five different residuals for further assessment of the model fit (i.e., "Resraw"=observed count - predicted (expected) count, "Reschi"=pearson residual, "Resdev"= deviance residuals, "StReschi"=standardized pearson residuals, etc.)

Full Table