How to explain the large p.value difference between these two models I fit two gee models with two different correlation structures, exchangeable vs ar(1), which resulted in very different p.values. I'm wondering what reasons have led that. Would somebody offer an explanation?
> summary(fit1)

Call:
geeglm(formula = LAU ~ SAMPLENO, data = bd, id = MAGE, corstr = "exchangeable")

 Coefficients:
             Estimate   Std.err   Wald Pr(>|W|)    
(Intercept)  0.067349  0.004537 220.32   <2e-16 ***
SAMPLENO    -0.002800  0.000947   8.74   0.0031 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters:
            Estimate  Std.err
(Intercept) 0.000425 0.000166

Correlation: Structure = exchangeable  Link = identity 

Estimated Correlation Parameters:
      Estimate Std.err
alpha    0.206   0.249
Number of clusters:   9   Maximum cluster size: 8 




> summary(fit2)

Call:
geeglm(formula = LAU ~ SAMPLENO, data = bd, id = MAGE, corstr = "ar1")

 Coefficients:
            Estimate  Std.err   Wald Pr(>|W|)    
(Intercept)  0.06392  0.00588 118.25   <2e-16 ***
SAMPLENO    -0.00142  0.00219   0.42     0.52    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters:
            Estimate  Std.err
(Intercept) 0.000427 0.000172

Correlation: Structure = ar1  Link = identity 

Estimated Correlation Parameters:
      Estimate Std.err
alpha    0.659   0.221
Number of clusters:   9   Maximum cluster size: 8

 A: It is a very good question. The correlation structures exchangeable and ar1 may “explain” different amounts of the variance of the response variable LAU, which will justify the need for additional predictors or not.
I will try to give an intuitive example. Imagine that LAU is a pure AR(1) process. In this case, the predictor SAMPLENO is useless and can be dropped from the model, so you would expect to obtain a non-significant p-value when testing for the nullity of this term in the second model.
Things would be different when using the first model. In this case, the exchangeable correlation structure may be a poor description, because the correlation between two different observations is assumed to be a constant that does not depend on their time of occurrence (and this assumption is false in the case of AR(1) processes). This mis-specification would give the wrong error model for the residues, and most likely would give an underestimate of their variance upon fitting the model. This in turn would explain why SAMPLENO is suddenly associated with a significant p-value. The time-dependence that could not be captured by the exchangeable model would now leak into this predictor (which seems to be a discrete time as the name suggests).
Something similar sometimes happens in the simpler situation of a $t$-test. A key assumption there is that sampling is IID, but if there is some correlation between the values (you could imagine that they are taken from an AR(1) process for instance), the usual formula for the variance is wrong and gives a too low estimate. As a result, the expected distribution of the mean is too narrow, which means that the null hypothesis is rejected too often.
In all those cases, remember that rejecting the null hypothesis means that something seems to be wrong with it. But this something could be any of the assumptions you made, including


*

*the distribution of the observation

*the correlation structure between them

*the nullity of a parameter.


In short, the p-value of the first model is significant probably because the exchangeable correlation structure is a poor description, not because SAMPLENO is a good predictor.
