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I know that Generalized Additive Model is used in Many Health Science Analysis. However, I wonder why it only consider some weather variables and time variable enter image description hereas independent variable not other factors such as car emission, number of factory.. etc that might effect dependent variable (such as number of the death cause by respiratory disease). Can some one explain about this ? the picture is example of gam that often use in Health Science analysis. Can someone explain why it only consider these the independent variables? not consider other factors ?

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    $\begingroup$ The emissions from cars and factories lead to PM10s and the amount of PM10s in the air is (as I understand it) the (hypothesised) proximal cause of some deaths from respiratory problems. So you might want a model with PM10s as the response to try to understand better the conditions that give rise to dangerous levels of particulates in the air. But if the model is to investigate to what extent PM10s can explain deaths due to respiratory problems, it makes sense to include the hypothesised proximal causal mechansim as the key term in the model. $\endgroup$ – Reinstate Monica - G. Simpson Sep 28 '18 at 16:44
  • $\begingroup$ Please write a more informative title. $\endgroup$ – Franck Dernoncourt Sep 29 '18 at 0:11
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Let's say Y denotes the number of deaths due to respiratory problems, measured each day for a period of time in a particular city (e.g., Mexico City). Your primary interest is in uncovering the association between Y and PM10.

To keep things simple for the purpose of this explanation, you can fit a simple linear regression model of the form:

Y = beta0 + beta1 * PM10 + error 

to the data. If you plot the residuals for this model, you are going to see that they display strong temporal features (including seasonality), which need to be accounted for in your model. To this end, you will expand the model to include a day_of_week term and a day_number term.

But when you plot the residuals from this new model against climate variables, there will still be evidence of systematic patterns in the plot of residuals vs. temperature and the plot of residuals vs. humidity. So you'll throw the temperature and humidity terms in the model, allowing each of them to have potentially non-linear effects.

The plot of residuals versus fitted values for this last model will likely look fine - just randomly scattered points about the zero line. Same for the plots of residuals vs. the time and climate predictors.

So controlling for temporal and climate variables when trying to estimate the association between Y and PM10 will result in well-behaved residuals, which will facilitate inferences about this association: Is the association significant? How big is the association?

The effect of car emissions, factory pollution, etc., is likely captured by PM10. Unless you want to estimate explicitly how car emission levels or factory pollution levels affect the number of deaths due to respiratory problems, you can think of PM10 as a "catch all" variable which will collect together all of these polluting influences.

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