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I am using logistic regression to model the effect of two production variables on the occurrence of tuberculosis. My independent variables are A (farm size: 0 for small, 1 for large) and B (cattle co-occurrence: 0 for no cattle, 1 for cattle presence). Reference levels for the veriables are:

  • Farm size = 0
  • Cattle presence = 0

The final model includes the interaction term A x B. Before presenting my main question, please see model output below:

global <- glm(TB ~ Farm size + Cattle + Farm size * Cattle, data = datos, family = binomial)
summary(global)

Deviance Residuals: 
Min       1Q   Median       3Q      Max  
-2.2101  -0.9660   0.4265   0.7461   1.4046  

Coefficients:
Estimate Std. Error z value Pr(>|z|)    
(Intercept)             -0.5199     0.2692  -1.931 0.053481 .
Farm: small              2.8713     0.7873   3.647 0.000265 ***
Cattle: +                1.3953     0.5965   2.339 0.019324 *  
Farm size * Cattle      -2.3253     1.0394  -2.237 0.025280 *  
 ---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)

Null deviance: 179.00  on 134  degrees of freedom
Residual deviance: 147.59  on 131  degrees of freedom

AIC: 155.59

Number of Fisher Scoring iterations: 4

Here, because of the presence of interaction, the main effects don't have a direct interpretation. Rather, we have to interpret them in the context of such phenomenon. At least in epidemiology, we work a lot using odds ratios by exponentiating variables' coefficients, and their 95% CI using the formula:

Upper CI = exp(coefficient + z * SE) (z being 1.96)
Lower CI = exp(coefficient - z * SE)

However, the calculation of odds ratios and their CI for the interaction term is not as straightforward. Following Hosmer and Lemeshow's book Applied Logistic Regression (2nd Ed., page 74, section 3.7), estimation of odds ratio in the presence of interaction should be derived from:

enter image description here

In our case, then, OR is calculated as follows:

Here, f represents Farm size, then f1 = 1 and f0 = 0. Also, x represents Cattle

OR <- round(exp(2.8713+(-2.3253)), digits = 2)
print(paste("OR interaction =", OR))  

"OR interaction = 1.73"

This OR represents the farm size effect (small - large) in farms with cattle, versus the farm size effect in farms without cattle. Or similarly, the odds of TB in small farms with cattle is 1.73 (73%) times the odds in small farms with no cattle.

The calculation of 95% CI for the interaction term is my main issue. Instead of using the term SE (1.0394, as calculated in the main model above), we need to use a variance-covariance matrix of the model and proceed as follows:

varcovar <- vcov(global)
varcovar

enter image description here

Using this matrix, the 95% CI for the interaction term's OR is calculated as follows according to the above authors (page 76, same book) and others:

enter image description here

Replacing matix values in equation 3.15 above, we obtain the variance, from which the SE can be derived:

SE <- sqrt(0.619774148 + 1.080447013 * 1 + (2 * (-0.619774148)))  
SE
[1] 0.6787289

Lastly, we use the estimated SE to derive 95% CI for the interaction term:

OR_lower_CI <- exp(0.546 - 1.96 * SE) 
OR_lower_CI
[1] 0.4564352    

OR_upper_CI <- exp(0.546 + 1.96 * SE)
OR_upper_CI
[1] 6.529358

The interaction term of the model is highly significant (p = 0.025280), and the 95% CI for the OR should not include 1.

¿How can this be possible? One would expect the CI not to include 1 (e.g., OR: 1.73; 95% CI: 1.4-6.5).

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Here, because of the presence of interaction, the main effects don't have a direct interpretation.

That's not true. The "Farm Small" effect is the log odds ratio for TB where there are 0 cattle (if it's a projection, shift "Cattle" to a useful center). The "cattle" effect is the odds ratio for "TB" for "one differing value of cattle" where farmsize is big. Those interpretations are ironic because you don't have data dictionary, so use your knowledge of the codes or design to refine as needed.

The Hosmer and Lemeshow equation for getting an OR for effect A conditional on effect B where A interacts with B is predicted at a specific value of the B (that's why setting the "B" to 0 leads to the interpretation above).

The reason why the predicted odds ratio of TB for Farm Size where Cattle = 1 is not statistically significant is that for that particular value of Cattle, the odds ratio you find is very close to 1. It's not surprising because the interaction is so significant. In fact, in any logistic regression model where there is a non-zero interaction term, you can project some value of the "B" variable to get an instantaneous odds ratio for A that is exactly the null hypothesis. It just so happens that this specific prediction of Cattle = 1 is within a range where the can't be excluded for that value. It's just a matter of correctly interpreting the output.

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