Solved – interpreting poisson regression models with log transformation and factors/qualitative variable

generalized linear modelinterpretationpoisson-regressionregression

I have a dataset with both quantitative ($x_1,x_2, \text{and} \ x_3$) and qualitative variables ($x_4$ – 4 levels ~0,1,2,3). 3 variables ($x_1,x_2,x_3$) have been log transformed. I do not know how to interpret coefficients when its log transformed.

glm(formula = y ~ log(1 + x1) + log(1 + x2) +                   
      log(1 + x3) + factor(x4), family = "quasipoisson",data = data)

                Estimate
(Intercept)     0.20
log(1 + x1)     0.76
log(1 + x2)     -0.1
log(1 + x3)     0.25
factor(x4)1     0.4
factor(x4)2     0.45
factor(x4)3     0.57

Let's suppose, if I want $x_4$ (for levels 0,1,2,3) to vary $x_1$ from 0,1,2,…,40 how would it effect my response considering everything being equal ? In addition, how to interpret $x_1,x_2, \text{ and } x_3$ ?.

Numerical Example, I want to vary $x_3$ between 0,1,2,3,4,5,… and so on and determine its impact on y for 4 different levels in variable $x_4$:

Let's suppose I want to predict for factor 0 which is when $x_4$ at 0 when $x_3 = 5$:

$$y = exp^{(0.20+0.25*5)}$$

Let's suppose I want to predict for factor 2 which is when $x_4$ at level 1 when x3 = 5:

$$y = exp^{(0.25*5+0.45)}$$

is my interpretation correct ?

Best Answer

Interpreting the coefficient of a log-transformed variables is reasonably straightforward: it represents the predicted change in the dependent variable for a 1-log-unit change in the independent variable.

Here, the dependent variable (in the default log-link for the quasipoisson family in glm) is $log(y)$. After the transformations of the variables $x_1$ through $x_3$, they are no longer the independent variables for the regression. The regression coefficients need to be interpreted in terms of the new independent variables $log(1+x_1)$ through $log(1+x_3)$.

So for the relation of $x_1$ to $y$, with the other independent variables held constant, you have a change of 1 log unit in $(1+x_1)$ corresponding to a change of 0.76 in $log(y)$. That pesky 1 in $log(1+x_1)$ makes is hard to provide a more general direct relation between $x_1$ itself and $y$.

Related Solutions

Solved – Missing factor levels after logistic regression glm()

This line in glm() is doing you in:

mf$drop.unused.levels <- TRUE

which is effectively setting the argument of the same name of model.frame(), which results in the behaviour you report.

The obvious solution is to not allow this to happen, to adjust your split sampling algorithm you use to produce your training and test sets. Instead of randomly sampling the rows of the data randomly sample within the levels of the factor.

If you don't want to handle the details yourself, try the caret package and its function createFolds():

## install.packages("caret")
library("caret")

X1 <- factor(rep(1:3, times = c(20, 30, 50))) ## dummy data for illustration
f <- createFolds(X1, k = 5)
f

which gives:

> f <- createFolds(X1, k = 5)
> f
$Fold1
 [1]  5  7 10 20 21 24 29 31 34 42 51 52 59 68 75 76 82 83
[19] 85 94

$Fold2
 [1]  4  9 11 18 22 23 30 38 40 44 55 58 62 66 70 72 80 81
[19] 87 92

$Fold3
 [1]  1 12 14 16 27 37 41 48 49 50 53 60 61 63 64 74 79 88
[19] 89 97

$Fold4
 [1]  3 15 17 19 25 28 32 35 36 43 54 57 67 69 71 73 78 86
[19] 98 99

$Fold5
 [1]   2   6   8  13  26  33  39  45  46  47  56  65  77  84
[15]  90  91  93  95  96 100

The values in f are the indices of the elements of X1 partitioning it into k = 5 groups, with sampling from within the levels of X1 as needed. Then take 1 of these folds at random as the test set.

## number of samples in levels of X1 for each split
> table(X1[-f[[1]]])

 1  2  3 
16 24 40 
> table(X1[-f[[2]]])

 1  2  3 
16 24 40 
> table(X1[-f[[3]]])

 1  2  3 
16 24 40 
> table(X1[-f[[4]]])

 1  2  3 
16 24 40 
> table(X1[-f[[5]]])

 1  2  3 
16 24 40

Do note that this algorithm doesn't guarantee that for small sample sizes that the stratified sampling will always work (i.e you may not be able to escape the missing levels issue in all cases).

Solved – Plotting linear regression with factors

When you use dummy variables, the coefficients don't represent slopes, they represent a constant number which is added to the estimate when the variable equals 1.

So your "High income: OECD" results from the linear regression are entirely consistent with the graph-- you can see on the graph that the High income: OECD line runs almost horizontally at about CO2 = 9, compared to your linear regression result of 8.7926.

If I understand ggplot correctly, it's plotting a separate regression for each income level. (A regression of CO2 levels on GDP.) So that's what you'd have to do if you want to get the same results as displayed on the graph.

As for your linear regression design, GDP will likely have some strange interactions with the income factors that will make the results difficult to interpret.

If the income factors are based on GDP per capita, GDP basically equals $$income factor \times population $$ Your results would be much clearer if you could run the regression with population instead of GDP. Then the interaction variables income_factor*population would make a lot of sense.

Best Answer

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Solved – Missing factor levels after logistic regression glm()

Solved – Plotting linear regression with factors

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