Measure correlation for distance and binary outcome

Question

I know enough about statistics to know that I don't know enough about statistics. This is therefore a humble question, and apologies in advance for asking a question that is likely to be missing key information or phrased badly.

I have two variables.

• One is a real-world metre distance between two points, and is therefore always >= 0. It doesn't seem to be normally distributed.
• The other is an outcome, with two values. It's not a 50:50 split.

I want to see if there is a correlation between these. Could someone advise me on a good way to do this?

A sample set of data is here. I'm providing this not out of laziness, but because it might be that there is some simple characteristic of this information which someone might want to know, and this might be quicker than asking me to provide it.

I would also be interested in the correct way to have phrased this question, or relevant tags.

Answer 1

First, note that your second column is a categorical variable, i.e. Yes/No, Up/Down in your case.

So, a first step will be to convert your non-binary data in the "Outcome" column to binary $0-1$ variables.

An Excel Implementation:

The next step will be to use the CORREL function in excel, to compute the correlation between your two types of variables - Distance and Outcome. To do so, apply =CORREL(A2:A3417,B2:B3417) in any empty cell (after you have converted your UP/Down into 1/0 respectively!) and you will obtain the correlation as desired.

R Implementation

(You can use readxl to read the .xlsx file directly instead)

If you try out the straightforward way in excel or by invoking cor(x,y) in R, you will find that there is almost no relation between the 2 variables, which kinda makes sense as one of them is a binary/categorical variable. ($cor(x,y) = -0.005750405$).

test <- read.csv("sample.csv", header = TRUE, sep = ',')
Outcome <- test$Outcome
Distance <- test$Distance
cor(Outcome,Distance)
-----------------------Outputs------------------------------
[1] -0.005750405

From the result above, I don't think that's the kind of result you want as not much can be interpreted from it. I highly suggest you take a look at Correlation between a nominal (IV) and a continuous (DV) variable, where a detailed breakdown of a problem similar to yours has been given.

In any case, if you are interested in descriptive statistics, you can try the following:

If you call a general linear model, you obtain the following:

lm1 <- glm(test$Outcome ~ test$Distance, data = test)
summary(lm1)
-----------------------Outputs------------------------------
Call:
glm(formula = test$Outcome ~ test$ï..Distance, data = test)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-0.8588   0.1413   0.1415   0.1417   0.1712  

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       8.588e-01  6.380e-03 134.606   <2e-16 ***
test$ï..Distance -4.717e-08  1.404e-07  -0.336    0.737    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.1218882)

    Null deviance: 416.14  on 3415  degrees of freedom
Residual deviance: 416.13  on 3414  degrees of freedom
AIC: 2508.7

Number of Fisher Scoring iterations: 2

Or if you call a linear model (lm):

lm1 <- lm(test$Outcome ~ test$Distance, data = test)
-----------------------Outputs------------------------------
Call:
lm(formula = test$Outcome ~ test$ï..Distance, data = test)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.8588  0.1413  0.1415  0.1417  0.1712 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       8.588e-01  6.380e-03 134.606   <2e-16 ***
test$ï..Distance -4.717e-08  1.404e-07  -0.336    0.737    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3491 on 3414 degrees of freedom
Multiple R-squared:  3.307e-05, Adjusted R-squared:  -0.0002598 
F-statistic: 0.1129 on 1 and 3414 DF,  p-value: 0.7369

Also remember that correlation coefficients are used in statistics to measure how strong a relationship is between two variables, not to show causation. More information about interpreting correlation can be found at http://www.statisticshowto.com/probability-and-statistics/correlation-coefficient-formula/ and https://www.dummies.com/education/math/statistics/how-to-interpret-a-correlation-coefficient-r/.

Answer 2

Welcome to crossvalidated. It looks like there is some "connection" between the two variables. Here are two histograms of the logarithm of the distance--one for each level of the Outcome variable. Observe that taking the log makes this variable look normally distributed.

If you regress log distance on outcome, there seems to be something going on. I get a p-value of $0.0254$ on the t-test. Hopefully this will give you some ideas. And here's the code used to generate all of this:

df <- read.csv("~/Desktop/sample_data_xv.csv")

# plot 1
hist(log(df$Distance[df$Outcome=="Up"]), col = "red", xlab="log distance", main = "log distance by outcome (red = up, blue = down)")
hist(log(df$Distance[df$Outcome=="Down"]), col = "blue", add = T)

# plot 2
hist(log(df$Distance[df$Outcome=="Up"]), col = "red", xlab="log distance", main = "log distance by outcome (red = up, blue = down)", freq = F)
hist(log(df$Distance[df$Outcome=="Down"]), col = "blue", add = T, freq = F)

# regress log distance on outcome
is.factor(df$Outcome)
mod <- lm(log(Distance+.000001) ~ Outcome, data = df)
summary(mod)

Edit: here's an image that plots density on the y-axis, instead of frequency, so it doesn't show the relative count as well. It does, however, show how the mean is shifted by the dummy variable in our regression.