rum <- read.csv("ruminationComplete.csv", na.string = "99") ## Imports the data into R
rum_scores <- rum %>% mutate(rumination = rowSums(across(CRQS1:CRSQ13)),
depression = rowSums(across(CDI1:CDI26))) ### I'm calculating
## total scores
corr <- cor(rum_scores$rumination, rum_scores$depression,
use = "pairwise.complete.obs") ## Correlation between rumination and depression
### Let's create a distribution of null correlations
nsim <- 100000
cor.c <- vector(mode = "numeric", length = nsim)
for(i in 1:nsim){
depre <- sample(rum_scores$depression,
212,
replace = TRUE)
rumia <- sample(rum_scores$rumination,
212,
replace = TRUE)
cor.c[i] <- cor(depre, rumia, use = "pairwise.complete.obs")
}
hist(cor.c, breaks = 120,
xlim= c(min(cor.c), 0.70),
main = "Histograma of null correlations")
abline(v = corr, col = "darkblue", lwd = 2, lty = 1)
abline(v = c(quantile(cor.c, .025),quantile(cor.c, .975) ),
col= "red",
lty = 2,
lwd = 2)
We can also do the same for the difference between means.
In this example, my null model is a model with null differences.
I also conducted permutations on this example.
We will estimate if the difference by sex in rumination is explained by chance.
In this sample the mean difference in rumination between males and females is \(\Delta M\) = 2.74.
set.seed(1236)
ob.diff <- mean(rum_scores$rumination[rum_scores$sex==0], na.rm = TRUE )- mean(rum_scores$rumination[rum_scores$sex==1], na.rm = TRUE)
### let's create a distribution of null differences
nsim <- 100000
diff.c <- vector(mode = "numeric", length = nsim)
### This is something called "loop", you don't have to pay attention to this.
for(i in 1:nsim){
women <- sample(rum_scores$rumination[rum_scores$sex==0],
length(rum_scores$rumination[rum_scores$sex==0]),
replace = TRUE)
men <- sample(rum_scores$rumination[rum_scores$sex==1],
length(rum_scores$rumination[rum_scores$sex == 1]),
replace = TRUE)
diff.c[i] <- mean(women, na.rm = TRUE)-mean(men, na.rm = TRUE)
}
hist(diff.c, breaks = 120,
#xlim= c(min(diff.c), 0.70),
main = "Histogram of Null Differences")
abline(v = ob.diff, col = "darkblue", lwd = 2, lty = 1)
abline(v = c(quantile(diff.c, .025), quantile(diff.c, .975) ),
col= "red",
lty = 2,
lwd = 2)
We can see how useful is a \(t\)-test by presenting a applied example.
In this example we will try to reject the null hypothesis that says:
“The rumination score in males is equal to the rumination score in females”
Also in this example I’m introducing a new function in R named t.test(). This is the function that will helps to know if we can reject the null hypothesis.
The function t.test() requires a formula created by using tilde ~.
In R the the variable on the right side of ~ is the independent variable, the variable on the left side of ~ is the dependent variable.
In a independent samples \(t\)-test the independent variable is always the group, and the dependent variable is always any continuous variable.
Two Sample t-test
data: rumination by sex
t = 2.2457, df = 203, p-value = 0.0258
alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
95 percent confidence interval:
0.3347849 5.1535481
sample estimates:
mean in group 0 mean in group 1
31.20896 28.46479 ***“IF we repeat the same analysis with multiple samples the probability of finding a*** \(t\)-value = 2.25 is p = 0.03, under the assumptions of the null model”.
I hope you remember JAMOVI. I mentioned this software in the first class.
It is also listed in the syllabus.
This is a software that runs R using a interface that does not require to write the R code. JAMOVI will write the R code for you behind scenes, it also runs the code for you. Pretty awesome isn’t it?
In this example I’ll generate 1000 “datasets” from a Gaussian process (normal distribution) with M = 60, sd = 100. Each data set will contain 10 observations.
After generating 1000 data sets, I’m estimating the mean of those simulated values.
Imagine you are asking the question how many minutes do you need to take a shower? to 10 participants, you then conduct the same study the next day with a different set of 10 participants every day until you get 1000 sets of 10 participants.
There is more beautiful qualities about the CLT. This rule applies to all distributions. When you increase the number of observations per sample, and the number of sample approach infinity. The sample of means will look Gaussian distributed (normally distributed).
Remember the Poisson distribution I mentioned some lectures behind?
As you can see the Poisson distribution after generating 1 data set of 10 observations doesn’t look like a Gaussian process at all!
Can the sample of means from a Poisson process generate a pretty bell shape?
In real life, you’ll never know the standard deviation of sample means, because this is a simulated example.
In real application you need to estimate how far your observed mean is from the true mean.
We can use this estimation \(\frac{\sigma}{\sqrt(n)}\) to approximate how far we are from the mean. In this formula \(\sigma\) is the standard deviation and \(n\) is the sample size.
The standard error decreases when we increase the sample size, let’s see another example:
Important
In simple words: The standard error (SE) measures the distance from the true value in the data generating process.
Westfall & Henning (2013):
