Meaning of this scatter plot

Question

I have two types of treatment, a and b, and I want to know if one is more painful than the other. I can apply both treatments to each patient, and ask for the pain experienced with each on a scale 0-10.

> head(df_long)
n_patient type pain
        1    a    2
        1    b    1
        2    a    7
        2    b    2
        3    a    1
        3    b    2

Here are the results for a survey with 20 patients.

I can test for normality of the distributions. I can test for significant differences in mean pain, etc. I can also make a scatter plot with points associated to patients, (x, y)=(pain_a, pain_b), which, it seems to me, will contain useful information.

However, I'm having trouble interpreting the plot. Patients have different pain tolerances. It seems natural that a patient giving a high pain score to treatment "a" will tend to rate highly the other one (at least for a certain range of pain similarity). In this plot, a positive correlation means that as patient's pain tolerance decreases, pain ratings for both treatments increase. A perfect correlation of 1 would mean that pain experienced in both treatments is the same. In my case, I obtain the following results:

      > cor.test(pain_a, pain_b,
                 alternative = "greater",
                 method = "pearson")

        Pearson's product-moment correlation

        data:  pain_a and pain_b
        t = 2.1551, df = 18, p-value = 0.02247
        alternative hypothesis: true correlation is greater than 0
        95 percent confidence interval:
          0.08914837 1.00000000
        sample estimates:
                      cor 
                 0.452883 

      > lm(b ~ a, df)

        Call:
        lm(formula = b ~ a, data = df)

        Coefficients:
        (Intercept)            a  
             1.0746       0.4102 

How do I read this r = 0.45? I would say that it's relevant that this r is smaller than 1, but I don't know the formal way of stating a conclusion out of this, or if this plot is just talking about pain tolerance in general, and it's not useful for distinguishing between treatments.

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EDITED

Following Bernhard's answer, one thing I can do is changing the basis of my problem, to look at the total pain experienced by the patient, $sum = a + b$, and the difference in pain, $diff = b - a$. Now variable $diff$ is restricted to the domain $[-sum, sum]$ for each $sum$ pain experienced by each patient, and the correlation and linear model that can be extracted from the scatter plot make much more sense.

For my invented example data though, I don't get any relevant results

> cor.test(pain_sum, pain_diff, method = "pearson")

    Pearson's product-moment correlation

data:  pain_sum and pain_diff
t = -0.47175, df = 18, p-value = 0.6428
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.5272479  0.3490807
sample estimates:
       cor 
-0.1105114 

> lm(diff ~ sum, df_sd)

Call:
lm(formula = diff ~ sum, data = df_sd)

Coefficients:
(Intercept)          sum  
     -1.050       -0.068  

But I think this is one useful thing to do in this type of scenarios. Thanks everyone for the answers!

Answer 1

The plot as such is useful; the linear regression and the correlation coefficient are not. You might instead consider drawing a line according to the function $y(x)=x$ of $a=b$ in the scatter plot so that you can see which dots are above and which below the line, thus indicating more or less pain in $b$. The $a=b$ line depicts equal pain in both conditions. Also, this plot has a lot of potential for overplotting (many points on top of each other). ggplot2 has a number of ways to address that. geom_count is but one example.

However, you might want to look into mean-difference plots, a.k.a. Bland-Altman plots for an often superior alternative to the simple scatter plot:

https://cran.r-project.org/web/packages/BlandAltmanLeh/vignettes/Intro.html

Edit after Edit of the question

Following my answer you came up with an edit, where you plotted the differences as a function of the sum (instead of the mean) of the pains and state, that you do not get any results from that. You may not get any results from correlation, but you get an interesting result from the plot: You can now easily see, that the difference between the pains is almost always negative, which means, that $b$ is associated with less pain than $a$. You can also see that this is true for probands with low pain scores as with probands with middle or high pain scores. The linear regression starts with a negative intercept at -1.05 and then drops even further with higher sums. You can conclude, that the prediction line is negative (meaning $a$ being more painfull) over the whole range of sums. That is some new information, we did not have before. You will need summary(lm(diff ~ sum, df_sd)) to see, whether this is significant. I cannot think of any value of the dashed lines through the origin. One of them follows the advice I gave for the original scatter plot, not for the mean-differences plot.

Again: You appear to have $20$ probands but only $17$ dots in your plot. This comes by overplotting and you should do something against it (ich geom_count is not to your taste, how about a sunflower plot?

Answer 2

Although the Pearson correlation coefficient $r$ can be great for assessing the direction and strength of the relationship between two variables (e.g. pain scores for treatment $a$ and for treatment $b$), it does not have as contextual of a meaning as the coefficient of determination $r^2$.

The coefficient of determination $r^2=.45^2=.2025$ tells us that $20.25$% of the variance in pain score for treatment $a$ can be accounted for by the pain score for treatment $b$ (and vice versa). This interpretation, in a way, may leave room for the influence of variables other than pain tolerance. Perhaps some of the variance in either pain score may be attributed to the impact/quality of the treatment.

Answer 3

Pearson correlation coefficient is a measure of the linear correlation between these two variables. Here the coefficient is equal to 0.45, so we do not observe a strong linear correlation between these two variables.

The sample size for running Pearson's r varies according to authors. In most cases it is written that a sample size equal or superior to 25 suffices. Sample size requirements for estimating pearson, kendall and spearman correlations