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This is a confusion sparked by problem 9.12 in Gelman and Hill (2007). An experiment on 50 cows was conducted to estimate effect of feed additive on milk fat. Four diets (treatments) were considered with different levels of additive (0.0, 0.1, 0.2, 0.3). Other covariates measured were lactation(seasons of lactation), age and initial weight of the cow. For the sake of argument lets assume that the cows were assigned to to treatments completely at random.

To facilitate meaningful interpretation of the regression coefficients, I centered the covariates.

cows$c_lactation <- (cows$lactation - mean(cows$lactation))
cows$c_initial_weight <- (cows$initial_weight - mean(cows$initial_weight))
cows$c_age <- (cows$age - mean(cows$age))

str(cows)
'data.frame':   50 obs. of  13 variables:
 $ level           : num  0 0 0 0 0 0 0 0 0 0 ...
 $ lactation       : int  3 3 2 2 2 1 1 1 3 3 ...
 $ age             : int  49 47 36 33 31 22 34 21 65 61 ...
 $ initial_weight  : int  1360 1498 1265 1190 1145 1035 1090 960 1495 1439 ...
 $ fat             : num  3.88 3.4 3.44 3.42 3.01 2.97 2.99 3.54 2.65 4.04 ...
 $ c_lactation     : num  0.62 0.62 -0.38 -0.38 -0.38 -1.38 -1.38 -1.38 0.62 0.62 ...
 $ c_initial_weight: num  101.94 239.94 6.94 -68.06 -113.06 ...
 $ c_age           : num  6.84 4.84 -6.16 -9.16 -11.16 ...

Then I run the following two linear models.

Model1:

summary(lm(fat ~ level + c_lactation + c_age + c_initial_weight, data = cows))

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)       3.2926488  0.1020850  32.254  < 2e-16 ***
level             1.8970077  0.5532560   3.429  0.00131 ** 
c_lactation       0.2840070  0.1693474   1.677  0.10046    
c_age            -0.0168047  0.0128543  -1.307  0.19775    
c_initial_weight  0.0003718  0.0005342   0.696  0.48999  

Residual standard error: 0.4204 on 45 degrees of freedom
Multiple R-squared:  0.3085,    Adjusted R-squared:  0.247 
F-statistic: 5.018 on 4 and 45 DF,  p-value: 0.001976

Model 2:

summary(lm(fat ~ factor(level) + c_lactation + c_age + c_initial_weight, data = cows))

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)       3.3351919  0.1254831  26.579  < 2e-16 ***
factor(level)0.1  0.1201635  0.1738695   0.691  0.49321    
factor(level)0.2  0.2722551  0.1750208   1.556  0.12714    
factor(level)0.3  0.5832468  0.1796321   3.247  0.00226 ** 
c_lactation       0.2880063  0.1722065   1.672  0.10170    
c_age            -0.0168912  0.0130611  -1.293  0.20283    
c_initial_weight  0.0002952  0.0005606   0.527  0.60113    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4268 on 43 degrees of freedom
Multiple R-squared:  0.3187,    Adjusted R-squared:  0.2236 
F-statistic: 3.352 on 6 and 43 DF,  p-value: 0.008469

contrasts(factor(cows$level))

    0.1 0.2 0.3
0     0   0   0
0.1   1   0   0
0.2   0   1   0
0.3   0   0   1

Questions:

  1. In interpreting the coefficient of level in model 1, is it correct to say that with a unit change in additive level (and other covariates at their mean) the milk fat level changes by 1.89 units? The intercept is the fat level without additive and at other covariates at their mean? Additive levels in this experiment vary only from 0.0 to 0.3, so is this a valid interpretation?

  2. How would I reconcile model 2 to model 1, given that some of the factor levels are now not significant?

  3. Why have the regression coefficients of the covariates changed by this conversion of the numeric factor to categorical (model 1 to model 2)?

  4. Why have the $R^2$ and the $F-$statistic changed (model 1 to model 2)?

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  1. You are right, but it is also important to mention that a unit change in additive level results in a 1.89 unit change in fat level assuming all other features are held constant.

  2. To compare the significance level, you will need to perform a type two analysis of variance test (ANOVA). Look at the package 'car' and do Anova(fit2). This groups all the categorical features and provides one p-value.

  3. All the regression coefficients will change because you have fundamentally fit a different model. Additive level treated as a continuous variable has a sense of magnitude. ex. 5>4>3>2>1. When treating it as a factor, there is no sense of magnitude. Therefore, 5 is not greater than 4. 5 and 4 are just different. You could change all the 5's so some letter 'foo' and it would not make any difference to the coefficients.

    This changes the model completely, and therefore all the coefficients will change.

  4. Because the models are again completely different, the $R^2$ and F-statistic will also change.

You are witnessing small changes in coefficient values. Sometimes, you might see dramatic changes and the signs may also change completely. Fit1 and Fit2 are fundamentally different models, and therefore you cannot compare the two. You can, however, test the accuracy of each of the models against a holdout set.

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