# R-squared equivalent for Generalized Estimating Equations (GEE) using a ordinal logistic regression model

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

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

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

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".

Zheng, B. (2000). Summarizing the goodness of fit of generalized linear models for longitudinal data. Statistics in Medicine, 19(10), 1265-1275. doi: 10.1002/(SICI)1097-0258(20000530)19:10<1265::AID-SIM486>3.0.CO;2-U

• @JRF1111 this isn't straight forward for me so I thought I'd try to write it out in R::geepack for a general & logistic (not at ordinal yet) GEE. Please tell me if I've understood you correctly. I'm kind of confused by the idea of running a second GEE to predict fitted values of the first. It seems like that should perform perfectly and in testing it has. Have I misunderstood something? I've posted the code @ github.com/bkellman/GEEreport/blob/master/reportFunctions.r Jul 21 '19 at 18:57
• @gung a related approach is to evaluate the predictive ability of a GEE. I’ve created a new R package doing just that using proper scoring rules: drizopoulos.github.io/cvGEE Jul 21 '19 at 20:51
• @DimitrisRizopoulos, it isn't my approach, that was a botched attempt to convert an answer to a comment. Hopefully, it's correct now. Jul 22 '19 at 0:55
• is it possible to use the second option for a continuse response variable?
– Moj
Dec 23 '19 at 16:35