Session4
Mandy
News
News
News
Ross says that that observing the global labour of love has tempered his cynicism.
Recap
Recap
You should know now:
- that statistics is all about simplifying
- we try to summarize and describe data through parameters
- location parameters (which? resp R commands?)
- scale parameters/parameters of spread (which? resp R commands?)
Recap
- mean(), median(), quantile()
- sd(), range(), IQR()
Recap
We have seen how
- a difference of means (of two groups)
- a spread parameter (standard deviation) and
the sample sizea (as a parameter of uncertainty)
are combined to measure the difference of the location of two groups this measure is compared to a null distribution through this comparison transformed in to a probability (p-value)
test statistic
\[t = \frac{\bar{X}_{male} - \bar{X}_{female}}{s_{overall}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}\]
Two sided alternative
One sided alternative - less
One sided alternative - greater
The p-value is the probability of the sample estimate (of the respective estimator - here our t) under the null. We do not know anything of probabilities of the null or of estimates under the alternative!
Decisions may be right - or wrong
| \(H_0\) is true | \(H_0\) is false | |
|---|---|---|
| \(H_0\) is not rejected | Correct decision | Type II error |
| \(H_0\) is rejected | Type I error | Correct decision |
T-tests in R
T-tests in R
- the one sample t-test
- the two sample t-test assuming equal variances
- the two sample t-test without the former assumption
- the paired t-test (in fact: this is a one sample t-test against 0)
- and there are others
But: there is only one command in R:
t.test()
One Sample t-test
set.seed(1)
x <- rnorm(12) ## create random numbers
t.test(x,mu=0) ## population mean 0
##
## One Sample t-test
##
## data: x
## t = 1.1478, df = 11, p-value = 0.2754
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## -0.2464740 0.7837494
## sample estimates:
## mean of x
## 0.2686377
One Sample t-test
t.test(x,mu=1) ## population mean 1
##
## One Sample t-test
##
## data: x
## t = -3.125, df = 11, p-value = 0.009664
## alternative hypothesis: true mean is not equal to 1
## 95 percent confidence interval:
## -0.2464740 0.7837494
## sample estimates:
## mean of x
## 0.2686377
Two Sample t-test
- we have given two numeric vectors
- we do not assume equal variance for the underlying distributions
set.seed(1)
x <- rnorm(12) ## create random numbers
y <- rnorm(12) ## create random numbers
head(data.frame(x,y))
## x y
## 1 -0.6264538 -0.62124058
## 2 0.1836433 -2.21469989
## 3 -0.8356286 1.12493092
## 4 1.5952808 -0.04493361
## 5 0.3295078 -0.01619026
## 6 -0.8204684 0.94383621
Two Sample t-test
t.test(x,y)
##
## Welch Two Sample t-test
##
## data: x and y
## t = 0.59393, df = 20.012, p-value = 0.5592
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.5966988 1.0717822
## sample estimates:
## mean of x mean of y
## 0.26863768 0.03109602
Two Sample t-test
- we have one numeric vector and one vector containing the group information
- we do not assume equal variance for the underlying distributions
g <- sample(c("A","B"),12,replace = T) ## create random group vector
head(data.frame(x,g))
## x g
## 1 -0.6264538 B
## 2 0.1836433 B
## 3 -0.8356286 A
## 4 1.5952808 B
## 5 0.3295078 A
## 6 -0.8204684 A
Two Sample t-test
t.test(x~g)
##
## Welch Two Sample t-test
##
## data: x by g
## t = -0.66442, df = 6.3524, p-value = 0.5298
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -1.6136329 0.9171702
## sample estimates:
## mean in group A mean in group B
## 0.1235413 0.4717726
Two Sample t-test
- we assume equal variance for the underlying distributions
t.test(x,y,var.equal = T)
##
## Two Sample t-test
##
## data: x and y
## t = 0.59393, df = 22, p-value = 0.5586
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.5918964 1.0669797
## sample estimates:
## mean of x mean of y
## 0.26863768 0.03109602
When should you use a t-test
- comparison of mean values against a null-value or against each other
- the t-test, especially the Welch test is appropriate whenever the underlying distributions are normal
- it is also recommended for group sizes \(\geq 30\) (robust against deviation from normality)
Exercises
Use the ALLBUS data set:
- do a t-test of income (V417): male against female (V81)!
- and compare the bmi (V279) in smokers and non-smokers (V272) and between people with high and normal blood pressure (V242)
- how would you interpret the results?
- visualize!
Rolling the dice
Rolling the dice
Suppose you are rolling a fair die 600 times: what is the number of 6s you are expecting?
And how many 6s do we need to reject the NULL (a fair die) using a two-sided test??
qbinom(p = c(0.025,0.975),size = 600, prob = 1/6)
## [1] 82 118
What do we have to change for a one-sided test?
Rolling the dice
Suppose you are rolling a fair die 600 times: what is the number of 6s you are expecting?
And how many 6s do we need to reject the NULL (a fair die) using a one-sided test on a 95% confidence level??
Alternative: greater
qbinom(p = 0.95,size = 600, prob = 1/6)
## [1] 115
Alternative: less
qbinom(p = 0.05,size = 600, prob = 1/6)
## [1] 85
Rolling the dice
So now let R rolling the dice!
sample(1:6,600,replace = T)
## [1] 2 3 4 6 2 6 6 4 4 1 2 2 5 3 5 3 5 6 3 5 6 2 4 1 2 3 1 3 6 3 3 4 3 2 5
## [36] 5 5 1 5 3 5 4 5 4 4 5 1 3 5 5 3 6 3 2 1 1 2 4 4 3 6 2 3 2 4 2 3 5 1 6
## [71] 3 6 3 3 3 6 6 3 5 6 3 5 3 2 5 2 5 1 2 1 2 1 4 6 5 5 3 3 5 4 4 3 2 6 4
## [106] 2 1 3 6 4 6 5 3 3 1 1 5 1 3 4 6 3 3 2 5 3 4 2 2 4 4 1 1 4 6 4 4 4 6 4
## [141] 5 4 2 2 5 3 2 5 1 6 4 4 2 3 4 2 4 1 2 2 2 6 3 5 6 3 1 3 5 3 4 6 6 3 3
## [176] 6 4 5 4 6 2 2 6 4 6 2 5 5 6 4 5 3 1 6 2 4 1 6 2 5 2 2 4 2 2 4 4 1 2 5
## [211] 6 1 5 6 5 2 4 6 6 3 2 1 2 4 6 4 2 1 3 6 3 1 3 4 3 5 5 4 3 3 2 4 6 1 3
## [246] 2 3 1 3 6 5 6 3 4 3 1 2 3 3 6 4 2 2 5 5 1 1 5 4 2 1 1 3 2 2 2 2 2 3 5
## [281] 1 4 6 3 1 1 2 1 1 2 2 1 6 2 4 5 1 1 1 6 5 1 3 3 3 6 2 5 1 3 1 2 6 5 2
## [316] 3 1 3 6 4 4 2 3 6 6 6 5 5 2 5 6 2 3 5 1 3 3 1 4 6 6 2 4 3 5 2 3 2 3 5
## [351] 3 1 5 3 6 2 6 1 6 3 1 2 6 2 4 6 1 5 5 3 3 4 6 2 2 3 3 6 2 3 3 4 4 6 2
## [386] 2 5 5 1 6 1 1 6 4 1 3 4 2 3 1 4 2 6 6 6 5 3 5 1 6 6 3 5 5 5 3 3 2 5 5
## [421] 3 4 4 1 3 3 1 6 6 6 6 1 3 4 1 5 5 1 3 6 1 2 4 4 2 3 3 6 1 1 6 3 3 2 1
## [456] 4 4 6 4 1 1 3 1 3 2 6 5 1 3 5 3 6 5 3 1 2 6 3 3 5 5 1 6 6 4 4 2 1 6 1
## [491] 1 6 2 1 3 3 3 4 6 1 4 5 4 4 3 6 3 6 5 4 6 3 1 5 5 2 4 4 3 4 5 5 3 1 3
## [526] 6 5 5 6 6 4 1 3 2 1 1 3 3 2 4 3 6 4 2 2 1 6 3 4 5 6 4 1 6 5 3 1 1 5 5
## [561] 4 1 3 5 6 5 2 4 2 6 1 3 2 1 5 4 5 3 6 5 3 6 4 1 4 3 1 4 6 5 4 2 5 4 5
## [596] 4 3 3 3 3
Rolling the dice
Count the 6s!
sample(1:6,600,replace = T)==6
## [1] FALSE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE
## [12] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE
## [23] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [34] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [45] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [56] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [67] FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE
## [78] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [89] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [100] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE
## [111] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [122] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [144] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [155] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE
## [166] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE
## [177] FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE
## [188] FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE
## [199] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [210] FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE TRUE TRUE FALSE
## [221] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE
## [232] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [243] TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE
## [254] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [265] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [276] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [287] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [298] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [309] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE
## [320] FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
## [331] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE
## [342] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [353] FALSE FALSE TRUE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE
## [364] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [375] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [386] FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE
## [397] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE
## [408] FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [419] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE
## [430] TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [441] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE
## [452] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [463] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [474] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE
## [485] FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE
## [496] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [507] FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [518] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [529] TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [540] FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [551] TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [562] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [573] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE
## [584] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [595] FALSE FALSE FALSE FALSE FALSE FALSE
Rolling the dice
Count the 6s!
sum(sample(1:6,600,replace = T)==6)
## [1] 100
Exercise
Now use the following code to replicate the experiment (rolling one fair dice 600 times) 1000 times. The number of 6s are stored in the vector x.
How many statistically significant results do you expect for a one-sided alternative? How many for a two-sided alternative?
How many statistically significant results did you get? (you can use table() in combination with a logical function)
Visualize the result using ggplot2 and geom_histogram() (look at the help of geom_histogram )!
x <- replicate(1000, sum(sample(1:6,600,replace = T)==6))
df <- data.frame(repid = 1:1000, n.6s = x)
Exercise
head(df,10)
## repid n.6s
## 1 1 98
## 2 2 110
## 3 3 92
## 4 4 103
## 5 5 115
## 6 6 105
## 7 7 108
## 8 8 107
## 9 9 105
## 10 10 118