Is there a measure that shows how well GEE using a ordinal logistic regression model explains the amount of variance in the data?

#### Best Answer

Five years late, but yes (kind of). Zheng proposed two $R^2$ analogues for GEE in 2000 (citation at bottom of answer).

### Option 1

For your ordinal logistic model, assume that there is an underlying continuous latent variable that, when thresholds are applied, results in your observed ordinal $Y$. (Also assume that your software allows you to access that latent variable.)

Run a second GEE predicting that latent variable with the same predictors used in the ordinal model. From there, you can use Zheng's marginal $R^2$:

$R_{marginal}^2 = 1- frac{sum_{c=1}^C sum_{i=1}^N (Y_{ic} – widehat {Y_{it}})^2} {sum_{c=1}^C sum_{i=1}^N (Y_{ic} – bar Y)^2} $

where the numerator is the sum of the squares of the Y (your latent variable) minus the fitted values from this second GEE across each cluster ( $c_1, c_2, … c_C$ ) and each observation ($i_1, i_2, … i_N$ ), and the denominator is the sum of the squares of the Y (your latent variable) minus the marginal mean of that Y.

### Option 2

Ignore the ordered nature of your outcome variable and use Zheng's $H_{marginal}$ as a measure of "proportional reduction in entropy due to the model" where your model becomes a multinomial logistic model. $H_{marginal}$ is defined as

$H_{marginal} = 1 – frac{sum_{c=1}^C sum_{i=1}^N sum_{k=1}^K hat pi_{cik} log(hat pi_{cik}) } { nTsum_{k=1}^K hat alpha_k log(hat alpha_k) } $

where $ pi_{ck} = P( Y_c = k | X) $ is the "model-based probability that a categorical response [in cluster $c$] equals $k$", $alpha_k = P(Y = k)$ is "the marginal probability of response $k$", and hats (^) indicate estimates.

Note that for both $R_{marginal}$ and $H_{marginal}$, you can obtain a "negative value when there is greater uncertainty in prediction under the model of than under the null model".