# 7.2.2 - Overdispersion

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Overdispersion is an important concept in the analysis of discrete data. Many a time data admit more variability than expected under the assumed distribution. The greater variability than predicted by the generalized linear model random component reflects overdispersion. Overdispersion occurs because the mean and variance components of a GLM are related and depends on the same parameter that is being predicted through the independent vector.

There is no such thing as overdispersion in ordinary linear regression. In a linear regression model

$y_i \sim N(x_i^T \beta, \sigma^2)$

the variance σ2 is estimated independently of the mean function $x_i^T \beta$.

With discrete response variables, however, the possibility for overdispersion exists because the commonly used distributions specify particular relationships between the variance and the mean; we will see the same holds for Poisson.

For the binomial response, if yi ~ Bin(ni, πi), the mean is μi = ni πi and the variance is μi(ni - μi) / ni.

• Overdispersion means that the data show evidence that the variance of the response yi is greater than μi(ni - μi) / ni.
• Underdispersion is also theoretically possible, but rare in practice. McCullagh and Nelder (1989) say that overdispersion is the rule rather than the exception.

In the context of logistic regression, overdispersion occurs when the discrepancies between the observed responses yi and their predicted values $\hat{\mu}_i=n_i \hat{\pi}_i$ are larger than what the binomial model would predict. Overdispersion arises when the ni Bernoulli trials that are summarized in a line of the dataset are

• not identically distributed (i.e. the success probabilities vary from one trial to the next), or
• not independent (i.e. the outcome of one trial influences the outcomes of other trials).

In practice, it is impossible to distinguish non-identically distributed trials from non-independence; the two phenomena are intertwined.

Issue: If overdispersion is present in a dataset, the estimated standard errors and test statistics the overall goodness-of-fit will be distorted and adjustments must be made. When a logistic model fitted to n binomial proportions is satisfactory, the residual deviance has an approximate χ2 distribution with (np) degrees of freedom, where p is the number of unknown parameters in the fitted model. Since the expected value of a χ2 distribution is equal to its degree of freedom, it follows that the residual deviance for a well-fitting model should be approximately equal to its degrees of freedom. Equivalently, we may say that the mean deviance (deviance/df) should be close to one. Similarly, if the variance of the data is greater than that under under binomial sampling, the residual mean deviance is likely to be greater than 1.

The problem of overdispersion may also be confounded with the problem of omitted covariates. If we have included all the available covariates related to yi in our model and it still does not fit, it could be because our regression function $x_i^T \beta$ is incomplete. Or it could be due to overdispersion. Unless we collect more data, we cannot do anything about omitted covariates. But we can adjust for overdispersion.

The most popular method for adjusting for overdispersion comes from the theory of quasilikelihood. Quasilikelihood has come to play a very important role in modern statistics. It is the foundation of many methods that are thought to be "robust" (e.g. Generalized Estimating Equations (GEE) for longitudinal data) because they do not require specification of a full parametric model. For more details see Agresti (2007, Sec 9.2) or Agresti (2013, Sec 12.2).

In the quasilikelihood approach, we must first specify the "mean function" which determines how μi = E(yi) is related to the covariates. In the context of logistic regression, the mean function is

$\mu_i=n_i \text{expit}(x_i^T \beta)$,

which implies

$\text{log}\left(\dfrac{\pi_i}{1-\pi_i}\right)=x_i^T \beta$

Then we must specify the "variance function," which determines the relationship between the variance of the response variable and its mean. For a binomial model, the variance function is μi(ni - μi) / ni.

But to account for overdispersion, we will include another factor σ2 called the "scale parameter," so that

$V(y_i)=\sigma^2 \mu_i (n_i-\mu_i)/n_i$.

• If σ2 ≠ 1 then the model is not binomial; σ2 > 1 is called "overdispersion" and σ2 < 1 is called "underdispersion."
• If σ2 were known, we could obtain a consistent, asymptotically normal and efficient estimate for β by a quasi-scoring procedure, sometimes called "estimating equations." For the variance function shown above, the quasi-scoring procedure reduces to the Fisher scoring algorithm that we mentioned as a way to iteratively find ML estimates.

Note that no matter what σ2 is assumed to be, we get the same estimate for β. Therefore, this method for overdispersion does not change the estimate for β at all. However, the estimated covariance for $\hat{\beta}$ changes from

$\hat{V}(\hat{\beta})=(X^T W X)^{-1}$

to

$\hat{V}(\hat{\beta})=\sigma^2 (X^T W X)^{-1}$

That is, the estimated standard errors must be multiplied by the factor $\sigma=\sqrt{\sigma^2}$.

How do we estimate σ2?

McCullagh and Nelder (1989) recommend

$\hat{\sigma}^2=X^2/(N-P)$

where X2 is the usual Pearson goodness-of-fit statistic, N is the number of sample cases (number of rows in the dataset we are modeling), and p is the number of parameters.

• If the model holds, then X2/(N - p) is a consistent estimate for σ2 in the asymptotic sequence N → ∞ for fixed ni's.
• The deviance-based estimate G2/(N - p) does not have this consistency property and should not be used.

This is a reasonable way to estimate σ2 if the mean model $\mu_i=g(x_i^T \beta)$ holds. But if important covariates are omitted, then X2 tends to grow and the estimate for σ2 can be too large. For this reason, we will estimate σ2 under a maximal model, a model that includes all of the covariates we wish to consider.

The best way to estimate σ2 is to identify a rich model for μi and designate it to be the most complicated one that we are willing to consider. For example, if we have a large pool of potential covariates, we may take the maximal model to be the model that has every covariate included as a main effect. Or, if we have a smaller number of potential covariates, we decide to include all main effects along with two-way and perhaps even three-way interactions. But we must omit at least a few higher-order interactions, otherwise we will end up with a model that is saturated.

In an overdispersed model, we must also adjust our test statistics. The statistics X2 and G2 are adjusted by dividing them by σ2. That is, tests of nested models are carried out by comparing differences in the scaled Pearson statistic, ΔX22, or the scaled deviance, ΔG22 to a chisquare distribution with Δdf degrees of freedom.

If the data are overdispersed — that is, if

$V(y_i) \approx \sigma^2 n_i \pi_i (1-\pi_i)$

for a scale factor σ2 > 1 ,  then the residual plot may still resemble a horizontal band, but many of the residuals will tend to fall outside the ± 3 limits. In this case, the denominator of the Pearson residual will tend to understate the true variance of the yi, making the residuals larger. If the plot looks like a horizontal band but X2 and G2 indicate lack of fit, an adjustment for overdispersion might be warranted. A warning about this, however: If the residuals tend to be too large, it doesn't necessarily mean that overdispersion is the cause. Large residuals may also be caused by omitted covariates. If some important covariates are omitted from xi, then the true πi's will depart from what your model predicts, causing the numerator of the Pearson residual to be larger than usual. That is, apparent overdispersion could also be an indication that your mean model needs additional covariates. If these additional covariates are not available in the dataset, however, then there's not much we can do about it; we may need to attribute it to overdispersion.

Note, there is no overdispersion for ungrouped data. McCullagh and Nelder (1989) point out that overdispersion is not possible if ni=1. If yi only takes values 0 and 1, the it must be distributed as Bernoulli(πi) and its variance must be πi(1 - πi). There is no other distribution with support {0,1}. Therefore, with ungrouped data, we should always assume scale=1 and not try to estimate a scale parameter and adjust for overdispersion.

### Summary of Adjusting for Overdispersion in the Binary Logistic Regression

The usual way to correct for overdispersion in a logit model is to assume that:

$E(y_i)=n_i \pi_i$
$V(y_i)=\sigma^2 n_i \pi_i (1-\pi_i)$

where σ2 is a scale parameter.

Under this modification, the Fisher-scoring procedure for estimating β does not change, but its estimated covariance matrix becomes σ2(XTWX)-1—that is, the usual standard errors are multiplied by the square root of σ2.

This will make the confidence intervals wider.

Let's get back to our example and refit the model, making an adjustment for overdispersion.

We can  refit the model, making an adjustment for overdispersion in SAS by changing the model statement to

model y/n= logconc / scale=pearson;

In SAS, including the option scale=Pearson in the model statement will perform the adjustment. It will not change the estimated coefficients βj, but it will adjust the standard errors.

Here's the output:

NOTE: The covariance matrix has been multiplied by the heterogeneity factor (Pearson Chi-Square / DF) 4.08043.

The estimated scale parameter is $\hat{\sigma}^2=X^2/df=4.08$. SAS automatically scales the covariance matrix by this factor, which means that

• the standard errors in the table of coefficients are multiplied by $\sqrt{4.08} \approx 2$, and
• the Wald statistics are divided by 4.08.

If you are using glm() in R, and want to refit the model adjusting for overdispersion one way of doing it is to use summary.glm() function. For example fit the model using glm() and save the object as RESULT. By default dispersion is equal to 1. This will perform the adjustment. It will not change the estimated coefficients βj, but it will adjust the standard errors.

Estimate from the MAXIMAL model dispersion value as X2/df. Then you can call

summary(RESULT, dispersion=4.08,correlation=TRUE,symbolic.cor = TRUE).

This should give you the same model but with adjusted covariance matrix, that is adjusted standard errors (SEs) for your beta's (estimated logistic regression coefficients) and also changed z-values. Notice it will not adjust overall fit statistics. For that try the package "dispmod" (see assay.R).

R output after adjusting for overdispersion:

There are other corrections that we could make. If we were constructing an analysis-of-deviance table, we would want to divide G2 and X2 by $\hat{\sigma}^2$ and use these scaled versions for comparing nested models. Moreover, in reporting residuals, it would be appropriate to modify the Pearson residuals to

$r_i^\ast=\dfrac{y_i-n_i\hat{\pi}_i}{\sqrt{\hat{\sigma}^2n_i\hat{\pi}_i(1-\hat{\pi}_i)}}$ ;

that is, we should divide the Pearson residuals (or the deviance residuals, for that matter) by $\sqrt{\hat{\sigma}^2}$.