What statistical test should I perform for the data who comes from more than one population? [closed]

I want to check the chosen property (for example volume, density etc.) between data of experiment 1 and experiment 2, whether the difference of property between the experiment is significant or not?

My experiment 1 consist of 4 samples and experiment 2 consist of 3 samples.

My data in each sample comes from spatial region and have following sizes:

Experiment 1

• sample 1: (300x1)
• sample 2: (250x1)
• sample 3: (200x1)
• sample 4: (350x1)

Experiment 2

• sample 1: (800x1)
• sample 2: (750x1)
• sample 3: (650x1)

What method will be good to perform statistical test.

(1) Merging all the samples of experiment 1 and experiment 2 and then perform the statistics.

(2) Merging all the samples of experiment 1 and experiment 2, plotting their histogram and perform the test on histogram value.

(3) Making similar bin (binsize 10) for each sample, take averages over bin for the samples in each experiment and then perform the statistical test.

I tried Kolmogorov-Smirnov for method 1; Spearman's rank correlation coefficient and Chi-square test for method 2 and 3 but none of p-values looks promising. I suspect that data in each samples comes from more than one population (meaning spatially different population) and these population may not be normally distributed.

What will you suggest: (i) what is the good way to arrange the data; and (2) which statistical should be done for such data?

• I'm not sure exactly what you mean by the $\times 1.$ Is the number $1$ subject to variation? A nonparametric test (such as two-sample Wilcoxon) will not be able to declare a significant difference at the 5% level, unless you supposed (before the experiments were done) that Expt 2 gives larger values than Expt 1. Sep 22 '19 at 16:23
• Welcome to CV, user3704712. I suspect your question will get better responses, and be less likely to be closed for being too unclear if you can edit "whether the difference of property between the experiment is significant or not?" to make explicit which difference(s) you care about? (e.g., mean? variance? distribution? etc.) Sep 22 '19 at 18:10
• @BruceET Sorry for the confusion. (300 x 1) means 300 rows in a column. Meaning length of data in sample 1 is 300. It is column vector. Sep 23 '19 at 6:17
• @Alexis Each element in each sample is an object, which has a property like mass, density, volume, surface area etc. I want to check whether volume of objects in each sample of experiment 1 is consistently different from volume of objects in each sample of experiment 2. Sep 23 '19 at 6:21
• Suspected something like that. My answer responds to that. Sep 23 '19 at 6:21

Two-sample nonparametric Wilcoxon test. The two-sample Wilcoxon rank sum test looks only at ranks (relative positions). It is true that all of the outcomes in one experiment are substantially greater than any of the outcomes in the other. However, there are only $${7 \choose 4}$$ possible ways for the $$3 + 4 = 7$$ outcomes to be allocated to two groups, and two of those show complete separation. And $$2/35 \approx 0.057 > 0.05.$$

Using data as shown below, here are results from 2-sample Wilcoxon tests as implemented in R. First, the two-sided test that the two groups are have different locations; second, the one-sided test that group 2 has smaller values than group 2.

x1 = c(300, 250, 200, 350)
x2 = c(800, 750, 650)
wilcox.test (x1, x2)        # two-sided test

Wilcoxon rank sum test

data:  x1 and x2
W = 0, p-value = 0.05714
alternative hypothesis:
true location shift is not equal to 0

wilcox.test(x1, x2, alt="l")

Wilcoxon rank sum test

data:  x1 and x2
W = 0, p-value = 0.02857
alternative hypothesis:
true location shift is less than 0


Welch two-sample t test. If you are willing to assume that your data are nearly normal, then you could use a t test. It is not feasible to test such small samples for normality, so the judgment whether to use a t test would have to be judged primarily on the basis of previous experience with such data. Results of both two-sided and one-sided tests give highly significant results with P-values much smaller than 0.05. [Welch two-sample t tests do not assume that the population variances are the same.]

t.test(x1, x2)

Welch Two Sample t-test

data:  x1 and x2
t = -8.3874, df = 3.9593, p-value = 0.001156
alternative hypothesis:
true difference in means is not equal to 0
95 percent confidence interval:
-610.6699 -305.9968
sample estimates:
mean of x mean of y
275.0000  733.3333

t.test(x1, x2, alt="less")

Welch Two Sample t-test

data:  x1 and x2
t = -8.3874, df = 3.9593, p-value = 0.0005779
alternative hypothesis:
true difference in means is less than 0
95 percent confidence interval:
-Inf -341.4931
sample estimates:
mean of x mean of y
275.0000  733.3333


Permutation test: Using difference of means as metric.

If your data are numeric in the sense that sample means are valid measures of centrality, then you could use a permutation test. This type of test takes numerical values into account, but does not require normal data.

At each step the data are scrambled between two 'groups' and the difference in means is found. The P-value of the two-sided test is about $$0.03 < 0.05.$$

set.seed(822)
d.obs = mean(x1) - mean(x2)
x = c(x1,x2)
m = 10^5;  d.prm = numeric(m)
for (i in 1:m) {
prm.x = sample(x)
d.prm[i] = mean(prm.x[1:4]) - mean(prm.x[5:7])
}
mean(abs(d.prm) >= abs(d.obs))
[1] 0.02871


A histogram of the simulated permutation distribution is shown below. The P-value is the probability below the vertical red line.

Using the pooled t statistic as metric. It may seem more familiar to use the pooled t statistic as the measure of difference in means between two groups. That the t statistic does not have Student's t distribution with $$\nu = n_1 + n_2 - 2$$ degrees of freedom (for our data $$\nu = 5,)$$ does not imply it's a poor measure.

In the program below, we permute the group labels instead of the data, but the effect is the same. Also the P-value of this permutation test turns out to be very similar to the P-value for the permutation test above.

x1 = c(300, 250, 200, 350);  x2 = c(800, 750, 650)
x = c(x1,x2); g = c(1,1,1,1,2,2,2)
t.obs = t.test(x~g, var.eq=T)$$stat set.seed(2019) t.prm = replicate(10^5, t.test(x~sample(g),var.eq=T)$$stat)
mean(abs(t.prm) >= abs(t.obs))
[1] 0.0279


The density curve of $$\mathsf{T}(\nu = 5)$$ is superimposed on the histogram of the simulated permutation distribution. Obviously, the permutation distribution is not a t distribution. Perhaps this casts doubt on using a pooled 2-sample t test to analyze these data. But the permutation distribution does not assume normality, and gives a useful result.

Ref: Wikipedia has a good explanation of permutation tests. Section 4 of this paper discusses two-sample permutation tests. Also, you can search this site for other examples of 'permutation tests'.