Weighted linear regression using the inverse of variance as the weighting factor

Question

I have the following data set that establishes a relationship between two variables "X" and "Y":

df <- data.frame(X=c(25,25,25,25,25,25,50,50,50,50,50,50,
75,75,75,75,75,75,100,100,100,100,100,100,
125,125,125,125,125,125,150,150,150,150,150,150),    
Y=c(2457524,2391693,2450828,2391252,2444638,2360293,
4693194,4844527,4835596,4878092,4809226,4722253,
7142763,7182769,7135550,7173920,7216871,7076359,
9496553,9537788,9405825,9439201,9609870,9707734,
12031958,12027037,11935594,11930086,12154132,
12096462,14298064,14396607,13964716,14221039,
14283992,14042220))

Consider the following problem:

"Adjust a weighted linear model using the "lm" function and, as the weighting factor, the inverse of the "Y" variance for each "X" level". That is, the linear model should be weighted by the inverse of the variance of each level of "X". In this case, how can we specify the weighted functional relation? Is there any specific function to be entered as argument in "Weights”?

Technical detail: Only fit by the function "lm". Not fit by any other method (gls, glm, etc.).

Answer 1

Just create the desired weight array to solve this problem. In your case, I called this vector pesos:

variancias_condicionais <- aggregate(df$Y, list(df$X), var)$x
quantidade_X <- as.numeric(table(df$X))
pesos <- rep(1/variancias_condicionais, quantidade_X)

ajuste <- lm(Y ~ X, data=df, weights=pesos)
summary(ajuste)

Call:
lm(formula = Y ~ X, data = df, weights = pesos)

Weighted Residuals:
     Min       1Q   Median       3Q      Max 
-2.17331 -0.71861 -0.08895  0.84733  2.42540 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    28185      22538   1.251     0.22    
X              95300        330 288.777   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.097 on 34 degrees of freedom
Multiple R-squared:  0.9996,    Adjusted R-squared:  0.9996 
F-statistic: 8.339e+04 on 1 and 34 DF,  p-value: < 2.2e-16