How to Do a Weighted T-Test in R?

Below is more like a small summary of my thoughts and quick search. I have never used a weighted t.test before, only weights in linear regression.

There is no clear definition for what would make a weighted t-test. The issue lies with how to use weights in estimating the error because that is the basis of your t-test. You can check out this discussion and maybe this paper on weights in linear regression.

So your data:

df = structure(list(PopDens = c(93.53455, 137.13861, 35.98619, 89.698, 
16.27796, 25.33346, 89.698, 46.27796, 25.33346, 36.27796, 1.33346
), Score1 = c(17.985288, 10.549394, 13.392857, 8.644537, 29.591635, 
21.081301, 2.644537, 29.591635, 5.081301, 29.591635, 9.081301
), Group = structure(c(2L, 1L, 1L, 2L, 1L, 4L, 3L, 1L, 2L, 1L, 
2L), .Label = c("A", "B", "C", "F"), class = "factor")), class = "data.frame", row.names = c(NA, 
-11L))

We subset on only A and B:

df = subset(df,Group %in% c("A","B"))

And we can compare the results of a t-test and lm:

coefficients(summary(lm(Score1~ Group,data=df)))
             Estimate Std. Error   t value     Pr(>|t|)
(Intercept)  22.54343   3.653195  6.170881 0.0004580837
GroupB      -12.34532   5.479793 -2.252882 0.0589470215

t.test(df$Score1[df$Group=="B"],df$Score1[df$Group=="A"],data=df)

    Welch Two Sample t-test

data:  df$Score1[df$Group == "B"] and df$Score1[df$Group == "A"]
t = -2.404, df = 6.463, p-value = 0.05007
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -24.695931765   0.005282865
sample estimates:
mean of x mean of y 
 10.19811  22.54343

You get a p-value of 0.0589470215 for the effect of difference of B from A. For the t.test 0.05007, it's not crazily different.

Now for a weighted linear regression:

coefficients(summary(lm(Score1~ Group,data=df,weight=df$PopDens)))
             Estimate Std. Error    t value   Pr(>|t|)
(Intercept) 17.845885   3.780246  4.7208269 0.00215547
GroupB      -5.466244   5.727617 -0.9543663 0.37168503

You can see that the coefficients are estimated differently.. more towards the higher weight samples.

For the weighted t-test offered in package weights:

library(weights)
wtd.t.test(x=df$Score1[df$Group=="A"],y=df$Score1[df$Group=="B"],
weight=df$Score1[df$Group=="A"],weighty=df$Score1[df$Group=="B"],samedata=FALSE)
$test
[1] "Two Sample Weighted T-Test (Welch)"

$coefficients
   t.value         df    p.value 
2.90701563 6.97938063 0.02283172 

$additional
Difference     Mean.x     Mean.y   Std. Err 
 13.468496  25.884728  12.416232   4.633101 

Apparently it is a frequency weight in this weighted t-test but I am not sure. If you prefer to use this, will be good to read the code in detail since it is not very well documented how the standard errors etc are calculated.

If you would have more than 2 groups, you could also do an wighted anova with:

library(stats)
aov(Score1 ~ Group, data = df1, weight = PopDens)