Multiple regression: Strength measure changes by student but is constant over time Disclaimer: This is the first time I use this forum, so please let me know if my question is not clear or if you need more information.
I have data which looks like this:

I want to run the regression:
lm(formula = SCORE ~ STRENGTH, data = mydata), in order to find out if higher STRENGTH of the student (NAME) leads to a higher test score. 
Note that the STRENGTH score changes by student but is constant over time. I was wondering if this regression setup is valid, or if the fact that I have no time variation is a problem in a time series regression? Would this allow me to test whether or not higher strength explains higher test scores?
Thank you very much for your help and suggestions.
Glenn
 A: The code lm(formula = SCORE ~ STRENGTH, data = mydata) assumes the data agrees to the following model ($Y=$ score and $X = $ strength)
$$Y_i = \beta_0 + \beta_1 \cdot x_i+\epsilon_i$$
where the $\epsilon_i$ are independent and $\mathrm{Var}[\epsilon_i] = \sigma^2$.
You would like to test whether $\beta_1 > 0$.
I would say that the assumptions seem unreasonable in your case, since the samples are not independent. I think the least-squares estimates are still unbiased but the variance estimate of $\hat \beta_1$ won't.
I think you could get an understanding of the issue by making the setting more extreme. Say you have 100 samples of Bob's strength and score, and one sample of Jim's strenght and score. If you ignore this fact and perform linear regression the point estimate of $\beta_1$ could be spot on but the variance estimate would be severly underestimated.
I've tried some code and I don't think you will be able to detect a significant difference (or increase)...
m <- matrix(c(0.81,0.67,0.98,0.23,0.17,0.38,0.49,0.11,0.73,0.27,0.51,0.3,0.5,0.8,0.5,0.5,0.8,0.3,0.3,0.5,0.3,0.8),ncol=2)
m <- as.data.frame(m)
m <- cbind.data.frame(m,c("M","J","B","J","J","B", "M", "M","J", "M","B"))
colnames(m) <- c("score","strength","name")
m

lm.fit <- lm(score~strength,data=m)
summary(lm.fit) #2 sided test unable to reject the null

