Group A twice as big as Group B, but no significant difference? I want to compare the means between two measurement methods, but the readings are non-normally distributed. Despite almost every "Y" measurement being more than twice the corresponding "X" measurement, my T-Tests aren't showing any statistically significant difference.
Fair warning I am not a statistician by any means.
This example shows my issue:
individual <- c("A", "B", "C", "D", "E", "F","A", "B", "C", "D", "E", "F")
measure <- c("X","X","X","X","X","X", "Y","Y","Y","Y","Y","Y")
value <- c("350", "15", "7", "2", "1", "1","800", "60", "20", "5", "3", "1")


df <- data.frame(individual, measure, value)
df$measure <- as.factor(df$measure)
df$value <- as.numeric(df$value)
 

TT <- t.test(df$value~df$measure)
TT

Gives the following output:
        Welch Two Sample t-test

data:  df$value by df$measure
t = -0.59885, df = 6.8666, p-value = 0.5685
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -424.4406  253.4406
sample estimates:
mean in group X mean in group Y 
       62.66667       148.16667 

I also tried an ANOVA, to account for the individual:
res.aov2 <- aov(value ~ measure + individual, data = df)
summary(res.aov2)

and get:
            Df Sum Sq Mean Sq F value Pr(>F)  
measure      1  21931   21931   1.363 0.2956  
individual   5 531109  106222   6.604 0.0294 *
Residuals    5  80423   16085                 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

I've also tried a Wilcoxon Signed- Rank
wilcox.test(df$grp1, df$grp2, paired = TRUE)

and got:
    Wilcoxon signed rank test with continuity correction

data:  df$grp1 and df$grp2
V = 15, p-value = 0.05906
alternative hypothesis: true location shift is not equal to 0

Warning message:
In wilcox.test.default(df$grp1, df$grp2, paired = TRUE) :
  cannot compute exact p-value with zeroes

I'm not really sure what else I can do. I would accept that they're not significant, but surely there's something in the fact that they're twice the size?
 A: Since you are talking about corresponding $X$ and $Y$ measurements, you should probably use a paired test. However, with the t.test, you used the non-paired version. Furthermore, the t.test presumes normally distributed measurements and your measurements don't look very normal.
Now, if you have paired data that is not necessarily normally distributed, the wilcox.test would be appropriate, provided we have independent pairs.
And, as you can see, the p-value of wilcox.test is very near 0.05, a common significance level. Note, that the last pair $X=1, Y=1$ cannot be used since this is a draw. That is what the warning in the wilcox.test output is about.
Finally, considering that you have only a small amount of data and their values are very far spread out, I would not expect a very significant result.
A: I disagree mildly with the advice to jump straight to Wilcoxon-Mann-Whitney as too pessimistic.
The substantive information that might inform an analysis that makes sense scientifically as well as statistically has been removed by the bland names measure and value, but the data example suggests working on logarithmic scale.
Here is a supportive graph of the ordered data on logarithmic scale (a quantile plot, or if you prefer an empirical cumulative distribution plot with axes reversed).

I would be happy to pursue a $t$ test on such a scale, or perhaps even better to use a generalized linear model with logarithmic link.
However, if zero is a possible value, you may need something different, but the idea of working on a transformed scale still seems a serious competitor. GLM with logarithmic link presumes only that means are positive, not that all values are.
