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I began with a maximal model which looked something like this:

Response ~ Predictor 1 + Predictor 2 + Predictor 3

I used backwards stepwise elimination and likeilhood ratio tests to then try and find the mininmal model. But I got a strange result for one of the predictor terms. It was insignificant (p=0.5), and plotting the data showed no obvious correlation with it and the response variable. But when I remove it from the model and do a likelihood ratio test comparing the model with the predictor to the model without it, this says that the increase in deviance is highly significant (p=1.752173e-48).

I'm really confused by this and was wondering whether anyone experienced could weigh in on how I should be intrepreting this.

The factor of interest is msize. Here are the outputs:

model_one<-lm(larvae~pop*carcass+msize+fsize+generation, data=bdata) 
model_two<-lm(larvae~pop*carcass+fsize+generation, data=bdata)

anova(model_one)
Analysis of Variance Table

Response: larvae
              Df Sum Sq Mean Sq F value    Pr(>F)    
pop            6    946   157.7  2.8336  0.009653 ** 
carcass        1   4742  4742.1 85.1937 < 2.2e-16 ***
msize          1      1     0.7  0.0120  0.912944    
fsize          1   2563  2563.2 46.0491 1.888e-11 ***
generation     4   3571   892.7 16.0372 9.117e-13 ***
pop:carcass    6   1600   266.7  4.7911 7.810e-05 ***
Residuals   1091  60728    55.7                      


anova(model_2)

Analysis of Variance Table

Response: larvae
              Df Sum Sq Mean Sq F value    Pr(>F)    
pop            6   1067   177.8  3.1920 0.0041225 ** 
carcass        1   4633  4632.9 83.1951 < 2.2e-16 ***
fsize          1   2371  2370.5 42.5682 1.031e-10 ***
generation     4   3480   870.1 15.6243 1.900e-12 ***
pop:carcass    6   1448   241.4  4.3343 0.0002473 ***
Residuals   1123  62537    55.7                      



Likelihood ratio test:
k2<-logLik(model_two)[1]       (= -3906.125)
k1<-logLik(model_one)[1]          (= -3799.075)
score<- -2*(k2-k1)                    (=214.0992)
pchisq(score, df=1, lower.tail=FALSE) (=1.752173e-48)
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    $\begingroup$ Well it is odd. If I am not making a mistake, deviance is $D = 2\times (\mathcal{l}_{saturated} - \mathcal{l}_{model})$ thus the increase in deviance corresponds to $2 \times (\mathcal{l}_{bigger model} - \mathcal{l}_{smaller model})$. This corresponds to the likelihood ratio test statistic on the parameter you add to go from the smaller model to the bigger... (again maybe I'm wrong...) Would you have more details to share ? $\endgroup$
    – Pohoua
    Commented Jun 24, 2020 at 15:09
  • $\begingroup$ Yes, I'm confused too - I've added outputs to the main question. $\endgroup$
    – user265883
    Commented Jun 24, 2020 at 15:44
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    $\begingroup$ Maybe because they are not using the same samples? The df_res is 1091 with msize in it, and 1123 (instead of 1092) when it's absent. $\endgroup$
    – Penguin_Knight
    Commented Jun 24, 2020 at 15:49
  • $\begingroup$ Aha, yes - that makes sense. There are a small number of datapoints in the dataset which have no value for msize and so they will have been removed from the second regression. Thank you. $\endgroup$
    – user265883
    Commented Jun 24, 2020 at 15:54

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