Solved – Finding Degree of Freedom from 2 Logit Models

logitself-study

Hi I have a question on finding DF in logit models. Say I have a table (all fictional numbers)

8 cities smoking vs heart disease

and say I am testing goodness of fit (deviance) between the fitted model and saturated model.

1) main effect model : $$Logit[\pi (x,y))] = \alpha + \beta_{x}^{smoking} + \beta_{y}^{city}$$

2) Saturated model: $$ Logit[\pi (x,y))] = \alpha + \beta_{x}^{smoking} + \beta_{y}^{city} + \beta_{xy}^{smoking,city}$$

How do I calculate the DF from these 2 models? (the answer is 7 from stats software, but I don't understand how to get the answer manually). DF=degree of freedom = difference in number of parameters from the 2 models.

thanks

Best Answer

I guess that with DF you mean the number of degrees of freedom of the test comparing either model with a null model containing only the constant.

In the first model you add 7 indicator variables for city (the 8th is the reference category and is thus left out) and 1 for smoking, so in total 8 degrees of freedom

In the second model you have 7 main effects of city and 1 main effect of smoking and 7 interaction effects, so in total 15 degrees of freedom.

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Given you have a non-significant direct effect, you actually don’t know whether the true coefficient of the direct effect is positive or negative, and however there is no evidence that it is not null. Let's first assume it is null then. My interpretation is that your main explanatory variable (let’s call it $X$) has only an effect on your outcome (let’s call it $Y$) through mediators (let’s say $M$). In short, all the effect of $X$ on $Y$ is through mediators. However, if you disregard the issue of statistical significance and just consider the coefficient values, you should conclude that $X$ has a negative direct effect on $Y$, but a positive overall effect, given its indirect effect is higher than its (negative) direct effect.

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Justification of the link function: A link function $g(\mu): (0,1) \rightarrow \mathbb{R}$ assures that all fitted values $\hat \mu = g^{-1}(x^\top \hat \beta)$ are always in $(0, 1)$. This may not matter that much in some applications, e.g., because the predictions or only evaluated in-sample or are not too close to 0 or 1. But it may matter in some applications and you typically do not know in advance whether it matters or not. Typical problems I have seen include: evaluating predictions new $x$ values that are (slightly) outside the range of the original learning sample or finding suitable starting values. For the latter consider:

library("betareg")
data("GasolineYield", package = "betareg")
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## Error in optim(par = start, fn = loglikfun, gr = if (temporary_control$use_gradient) gradfun else NULL,  : 
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But, of course, one can simply try both options and see whether problems with the identity link occur and/or whether it improves the fit of the model.

Interpretation of the parameters: I agree that interpreting parameters in models with link functions is more difficult than in models with an identity link and practitioners often get it wrong. However, I have also often seen misinterpretations of the parameters in linear probability models (binary regressions with identity link, typically by least squares). The assumption that marginal effects are constant cannot hold if predictions get close enough to 0 or 1 and one would need to be really careful. E.g., for an observation with $\hat \mu = 0.01$ an increase in $x$ cannot lead to a decrease of $\hat \mu$ of, say, $0.02$. But this is often treated very sloppily in those scenarios. Hence, I would argue that for a limited response model the parameters from any link function need to be interpreted carefully and might need some practice. My usual advice is therefore (as shown in the other discussion you linked in your question) to look at the effects for regressor configurations of interest. These are easier to interpret and often (but not always) rather similar (from a practical perspective) for different link functions.

Best Answer

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