How to find the p-value for two sets of data in R?

Question

New to R, and I have two data sets -- they have the same x-axis values, but the y-axis varies.

I'm trying to find the correlation between the two. When I use R to draw the ablines through the scatter plot, it gives me two lines-of-best-fit that seemingly makes one data set higher than the other -- but I'd really like to know the p-value between these two data sets to know the effect.

After looking it up, it seems like I should use t.test -- but I'm unsure how to run them against each other.

For example, if I run:

t.test(t1$xaxis,t1$yaxis1)
t.test(t2$xaxis,t2$yaxis2)

It gives me the right means of x and y (t1: 16.84, 88.58 and t2: 14.79, 86.14) -- but for the rest, I'm not sure:

t1: t = -43.8061, df = 105.994, p-value < 2.2e-16

t2: t = -60.1593, df = 232.742, p-value < 2.2e-16

Obviously the p-values given are (a) microscopic, and (b) I don't know how to make it tell me about the data sets relationship with each other -- and not individually.

Any help is greatly appreciated -- thanks!

Answer 1

Since you asked for it, here is how I understand your problem.

You have two groups of y values corresponding to identical x values. Here I assume that the relationship between y and x is linear. If it isn't you could transform your variables, use a non-linear model, an additive model, ...

First let's simulate some data since you don't provide any:

set.seed(42)
x <- 1:20
y1 <- 2.5 + 3 * x +rnorm(20)
y2 <- 4 + 2.5 * x +rnorm(20)

plot(y1~x, col="blue", ylab="y")
points(y2~x, col="red")
legend("topleft", legend=c("y1", "y2"), col=c("blue", "red"), pch=1)

Now, we want to know if the two samples differ. We can find out by fitting a model:

DF <- cbind(stack(cbind.data.frame(y1, y2)), x)
names(DF) <- c("y", "group", "x")

fit <- lm(y~x*group, data=DF)
summary(fit)

Call:
lm(formula = y ~ x * group, data = DF)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.2585 -0.4603 -0.1899  0.9008  2.2127 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.51769    0.55148   6.379 2.17e-07 ***
x            2.92136    0.04604  63.457  < 2e-16 ***
groupy2      0.67218    0.77991   0.862    0.394    
x:groupy2   -0.46525    0.06511  -7.146 2.11e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.187 on 36 degrees of freedom
Multiple R-squared:  0.9949,    Adjusted R-squared:  0.9945 
F-statistic:  2333 on 3 and 36 DF,  p-value: < 2.2e-16

The intercepts are not significantly different, but the slopes are. If group is a significant effect, we can test best by comparing with a model that doesn't consider group:

fit0 <- lm(y~x, data=DF)
anova(fit0, fit)

Analysis of Variance Table

Model 1: y ~ x
Model 2: y ~ x * group
  Res.Df     RSS Df Sum of Sq      F    Pr(>F)    
1     38 300.196                                  
2     36  50.738  2    249.46 88.498 1.267e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

As you see, the samples are different.