# Correlation analysis on a very small dataset

I am doing a research on (something like) the relationship between hospital's personnel job satisfaction(as individuals) and the overall performance score of the given hospital(as a whole). the number of hospitals on which I am conducting the research is 3 with each having a different number of personnel. So, what I had in mind as to the statistical test for the task was to investigate the correlation between the average job satisfaction of each hospital and its performance score(3 rows of data (n-hospitals=3)) using the Pearson correlation test. But I'm inclined to think that this isn't a good approach due to the small set of data, and I should use anova test instead. which approach should I adapt? Thanks in advance.

I don't think I would use the averages. I would use the individual data. Having different number of individuals per hospitals might not be problematic. ... But --- if I understand the question correctly --- with only three hospitals, you might as well use anova or something of a similar spirit.

EDIT: I just added a plot and some R code for a visual. Here, the hospitals have 10, 20, and 30 observations. Their means are 4, 5, and 9. And the Individual scores are centered on 5, 6, and 7, with equal standard deviations. So I did doctor the data both to be correlated (r ~ 0.5), and to be homoscedastic.

In this case, either the correlation analysis or the anova analysis should work fine, but they do ask different questions.

set.seed(sum(utf8ToInt("Sal")))

A = c(rnorm(10 ,5, 1))
B = c(rnorm(20 ,6, 1))
C = c(rnorm(30 ,7, 1))
Y = c(A, B, C)
X = c(rep(4, length(A)), rep(5, length(B)), rep(9, length(C)))

cor.test(~ X + Y)

### data:  X and Y
### t = 4.3634, df = 58, p-value = 5.332e-05
### sample estimates:
###       cor
### 0.4971262

Data = data.frame(Hospital=X, Individual=Y)

library(ggplot2)

ggplot(Data, aes(x=Hospital, y=Individual)) +
geom_point()


• Thank you. However, there is another problem which is the fact that my dataset doesn't have a normal distribution, does this prevent me from using averages and correlation analysis? Commented Nov 15, 2021 at 12:58
• And also, is it correct to say that, since the hospitals differ in performamce scores and anova(or kruskal-wallis) test did not reject the null hypothesis for job satisfaction, the job satisfaction doesn't affect the performance score? Commented Nov 15, 2021 at 13:31
• On 1) Well, Pearson correlation doesn't really require normal data. See top answer and comments here: https://stats.stackexchange.com/questions/3730/pearsons-or-spearmans-correlation-with-non-normal-data. Also, there is the possibility of using Spearman correlation. Averages can be used with non-normal data, though average may not be the best measure of location for very skewed data. Commented Nov 15, 2021 at 19:04
• On 2) I don't think I would move to the second part of that conclusion: the job satisfaction doesn't affect the performance score. a) Because that's not really what you tested. b) Because it seems to suggest that a lack of evidence is the same as evidence of an absence. Commented Nov 15, 2021 at 19:17

Any test will suffer from such a small sample size, as it will give you hardly any power to detect deviations from your null hypothesis. I suggest to not run a statistical test at all, but rather just to plot your data and say what you see.