Session 4
Mandy / Thomas
20 Januar, 2017
Recap
You should know:
- statistics is all about simplifying
- we try to summarize and describe data through parameters of:
- LOCATION
- e.g. mean, median, mode
- SCALE
- e.g. variance, standard deviation
- SPREAD
- e.g. minimum / maximum / range / quantile / IQR
- LOCATION
- the meaning of these parameters
- necessary R commands
Recap
We have seen how
- parameters of
- location of two groups (means)
- spread (standard deviation)
- uncertainty (sample size) incremental: true
- to measure a difference of the location in a standardized manner
- this measure is compared to a t-distribution relating to a so called Null-Hypotheses which one “hopefully” will be rejected to show an effect could exist
Recap
We have seen how
- this comparision is transformed to a propabiltiy (p-value) to get this result by random
- comparing this propability with a defined maximum propability for a random result (normally 5%) gives the opportunity to decide whether the effect could exist or not
- important is also the effect size itself
(e.g. is an effect of \( \mu_2 - \mu_1 = 101 - 100 = 1 \) relevant?)
Test Statistic
- some property of two groups (Men and Women) are measured.
- to compare their means, we apply the so called T-Test
- to do this we compute the so called T-Statistic:
\[ t=\frac{{\bar{X}_{m}}-{\bar{X}_{w}}}{s_{overall}\sqrt{\frac{1}{n_{m}}+\frac{1}{n_{w}}}} \]
Decisions can 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 |
Alternative of alternatives
| alternative | options |
|---|---|
| one sided test | less |
| greater | |
| two sided test | equal |
| —————- | —————- |
| one sample test | |
| two sample test | equal variances |
| not necessary equal variances | |
| ————– | —————- |
| unpaired | |
| paired |
Alternative of alternatives
- two sided - equal
Alternative of alternatives
- one sided - less
Alternative of alternatives
- one sided - greater
T-Tests in R
There are many options more but only one command in R:
t.test( )
T-Tests in R
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
T-Tests in R
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
T-Tests in R
Two Samples T-Test
- we have given a two numeric vectors
- we do not assume equal variances for the underlying distributions
set.seed( 1 )
x <- rnorm( 12 )
y <- rnorm( 12 )
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
T-Tests in R
Two Samples 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
T-Tests in R
Two Samples T-Test
- we have one numeric vector and one vector containing the group information
- we do not assume equal variances for the underlying distribution
## create random group vector
g <- sample( c( "A", "B" ), 12, replace = T )
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
T-Tests in R
Two Samples 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
T-Tests in R
Two Samples T-Test
- equal variances now
t.test( x ~ g, var.equal = T )
Two Sample t-test
data: x by g
t = -0.71719, df = 10, p-value = 0.4897
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-1.430111 0.733648
sample estimates:
mean in group A mean in group B
0.1235413 0.4717726
When should one use the t-test?
- comparision of mean values against a population 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 a group size equal or larger than 30 (robust against deviation from normality)
Exercise
Use the ALLBUS data set:
- do a test of income (V417) for the groups male and female (V81)!
- compare the bmi (V279) of smokers and non-smokers (V272)
- compare the bmi (V279) of people with high and normal blood pressure (V242)
- how would you interprete the results?
- visualize!
Simulations with R
Rolling the dice
Suppose you are rolling a fair dice 600 times!
- How many sixes would you expect?
- How many sixes do we need to reject the \( H_0 \)-Hypotheses using a two-sided test?
- test for EQUALITY
qbinom( p = c( .025, .975 ), size = 600, prob = 1 / 6 )
[1] 82 118
Simulations with R
What do we have to change for a one-sided test?
- test for LESS
qbinom( p = .05, size = 600, prob = 1 / 6 )
[1] 85
- test for GREATER
qbinom( p = .95, size = 600, prob = 1 / 6 )
[1] 115
Simulations with R
Now let's R roll the dice.
## paranthesis are for executing this row instantly
( dice.trials <- sample( 1 : 6, 600, replace = T ) )
[1] 6 2 3 2 4 2 3 5 1 6 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
[36] 5 3 3 5 4 4 3 2 6 4 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
[71] 4 1 1 4 6 4 4 4 6 4 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
[106] 3 1 3 5 3 4 6 6 3 3 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
[141] 2 2 4 2 2 4 4 1 2 5 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
[176] 5 5 4 3 3 2 4 6 1 3 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
[211] 1 1 3 2 2 2 2 2 3 5 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
[246] 6 2 5 1 3 1 2 6 5 2 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
[281] 6 2 4 3 5 2 3 2 3 5 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
[316] 3 3 6 2 3 3 4 4 6 2 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
[351] 6 3 5 5 5 3 3 2 5 5 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
[386] 3 3 6 1 1 6 3 3 2 1 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
[421] 5 1 6 6 4 4 2 1 6 1 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
[456] 2 4 4 3 4 5 5 3 1 3 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
[491] 6 4 1 6 5 3 1 1 5 5 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
[526] 3 1 4 6 5 4 2 5 4 5 4 3 3 3 3 5 6 1 5 6 6 3 3 6 1 6 3 4 4 5 3 1 3 4 1
[561] 6 2 4 1 2 4 2 4 3 6 6 3 2 5 3 2 1 1 5 5 6 2 6 2 5 2 1 6 5 6 5 6 3 3 4
[596] 1 2 4 5 6
Simulations with R
Find the sixes!
dice.trials == 6
[1] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
[12] TRUE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE FALSE
[23] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[34] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
[45] FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE
[56] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
[67] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
[78] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[89] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[100] FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
[111] FALSE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE
[122] FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
[133] FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
[144] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE
[155] FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE
[166] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
[177] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
[188] FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
[199] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[210] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[221] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[232] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
[243] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
[254] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE
[265] TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
[276] FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE
[287] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE
[298] FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE
[309] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE
[320] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE
[331] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[342] FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE
[353] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[364] FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE
[375] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
[386] FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
[397] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
[408] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE
[419] FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE
[430] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
[441] FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE
[452] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[463] FALSE FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE FALSE
[474] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
[485] FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE
[496] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
[507] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[518] FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
[529] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
[540] FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE
[551] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
[562] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE
[573] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE
[584] FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE TRUE FALSE FALSE
[595] FALSE FALSE FALSE FALSE FALSE TRUE
Simulations with R
Count them!
## length( dice.trials[ dice.trials == 6 ] ) ## one way
sum( dice.trials == 6 ) ## another way
[1] 107
Simulations with R
Now let's R roll the dice very very often!
Now use the following code to replicate the experiment (rolling one fair dice 600 times) 1000 times!
The number of sixes are stored in the vector dice.trials.1000.
- 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! Alternatively you can use hist( ).
Simulations with R
Now let's R roll the dice very very often!
dice.trials.1000 <- replicate( 1000, sum( sample( 1 : 6, 600, replace = T ) == 6 ) )
df <- data.frame( repl.count = 1 : 1000, sixes.count = dice.trials.1000 )
head( df )
repl.count sixes.count
1 1 96
2 2 109
3 3 93
4 4 99
5 5 122
6 6 105
Simulations with R
Now let's R roll the dice very very often!
table( df$sixes.count )
64 73 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92
1 1 2 4 1 5 5 4 8 11 12 13 17 26 27 15 30 38
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
40 46 47 37 42 42 35 39 36 43 34 44 36 29 34 23 25 24
111 112 113 114 115 116 117 118 119 120 121 122 126 127 130 131
21 16 15 17 16 6 6 8 5 3 3 3 2 1 1 1
quantile( df$sixes.count, probs = c( 0, .025, .05, .5, .95, .975, 1 ) )
0% 2.5% 5% 50% 95% 97.5% 100%
64 83 85 99 115 118 131
Simulations with R
Now let's R roll the dice very very often!
hist( df$sixes.count, breaks = 50 )
Simulations with R
Now let's R roll the dice very very often!
library( ggplot2 )
## ??geom_histogram
ggplot( df, aes( sixes.count ) ) + geom_histogram( binwidth = 1, col = '#1A1A18', fill = '#98BD0E' ) + theme_bw( )
