Classical Tests

Exercises

• load the data (file: session4data.rdata)
• make a new summary data frame (per subject and time) containing:
  • the number of trials
  • the number correct trials (absolute and relative)
  • the mean TTime and the standard deviation of TTime
  • the respective standard error of the mean
• keep the information about Sex and Age\_PRETEST
• make a plot with time on the x-axis and TTime on the y-axis showing the means and the 95\% confidence intervals (geom\_pointrange())
• add the number of trials and the percentage of correct ones using geom\_text()

Exercises - Solutions

• load the data (file: session4data.rdata)
• make a new summary data frame (per subject and time) containing:

   1 > sumdf <- data %>%
   2 +     group_by(Subject,Sex,Age_PRETEST,testid) %>%
   3 +     summarise(count=n(),
   4 +               n.corr = sum(Stim.Type=="hit"),
   5 +               perc.corr = n.corr/count,
   6 +               mean.ttime = mean(TTime),
   7 +               sd.ttime = sd(TTime),
   8 +               se.ttime = sd.ttime/sqrt(count))
   9 > head(sumdf)
  10 Subject Sex Age_PRETEST testid count n.corr perc.corr mean.ttime 
  11 1       1   f        3.11  test1    95     63 0.6631579   8621.674 
  12 2       1   f        3.11      1    60     32 0.5333333   9256.367 
  13 3       1   f        3.11      2    59     32 0.5423729   9704.712 
  14 4       1   f        3.11      3    60     38 0.6333333  14189.550 
  15 5       1   f        3.11      4    59     31 0.5254237  13049.831 
  16 6       1   f        3.11      5    59     33 0.5593220  14673.525 
  17 Variables not shown: sd.ttime se.ttime (dbl)  

four possible situations

Common symbols

Alternatives

The p-value is the probability of the sample estimate (of the respective estimator) under the null. The p-value is NOT the probability that the null is true.

Z-test for a population mean

The z-test is a something like a t-test (it is like you would know almost everything about the perfect conditions. It uses the normal distribution as test statistic and is therefore a good example. To investigate the significance of the difference between an assumed population mean

and a sample mean

• It is necessary that the population variance

is known.

• The test is accurate if the population is normally distributed. If the population is not normal, the test will still give an approximate guide.

Excercise

• Write a function which takes a vector, the population standard deviation and the population mean as arguments and which gives the Z score as result.
  • name the function ztest or my.z.test - not z.test because z.test is already used
  • set a default value for the population mean
• add a line to your function that allows you to process numeric vectors containing missing values!
• the function pnorm(Z) gives the probability of

. Change your function so that it has the p-value (for a two sided test) as result.

• now let the result be a named vector containing the estimated difference, Z, p and the n.

You can always test your function using simulated values: rnorm(100,mean=0) gives you a vector containing 100 normal distributed values with mean 0.

Z-test for a population mean - solution

Write a function which takes a vector, the population standard deviation and the population mean as arguments and which gives the Z score as result.

   1 > ztest <- function(x,x.sd,mu=0){
   2 +     sqrt(length(x)) * (mean(x)-mu)/x.sd
   3 + }
   4 > set.seed(1)
   5 > ztest(rnorm(100),x.sd = 1)
   6 [1] 1.088874

Add a line to your function that allows you to also process numeric vectors containing missing values!

   1 > ztest <- function(x,x.sd,mu=0){
   2 +     x <- x[!is.na(x)]
   3 +     if(length(x) < 3) stop("too few values in x")
   4 +     sqrt(length(x)) * (mean(x)-mu)/x.sd
   5 + }

The function pnorm(Z) gives the probability of

. Change your function so that it has the p-value (for a two sided test) as result.

   1 > ztest <- function(x,x.sd,mu=0){
   2 +     x <- x[!is.na(x)]
   3 +     if(length(x) < 3) stop("too few values in x")
   4 +     z <- sqrt(length(x)) * (mean(x)-mu)/x.sd
   5 +     2*pnorm(-abs(z))
   6 + }
   7 > set.seed(1)
   8 > ztest(rnorm(100),x.sd = 1)
   9 [1] 0.2762096

Now let the result be a named vector containing the estimated difference, Z, p and the n.

   1 > ztest <- function(x,x.sd,mu=0){
   2 +     x <- x[!is.na(x)]
   3 +     if(length(x) < 3) stop("too few values in x")
   4 +     est.diff <- mean(x)-mu
   5 +     z <- sqrt(length(x)) * (est.diff)/x.sd
   6 +     round(c(diff=est.diff,Z=z,pval=2*pnorm(-abs(z)),n=length(x)),4)
   7 + }
   8 > set.seed(1)
   9 > ztest(rnorm(100),x.sd = 1)
  10 diff        Z     pval        n 

Requirements

• Z-test for two population means (variances known and equal)
• Z-test for two population means (variances known and unequal)

To investigate the statistical significance of the difference between an assumed population mean

latex error! exitcode was 1 (signal 0), transscript follows:

This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_d63a06fef3c6ed6ce99f7934d79874bb705593d0_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
No file latex_d63a06fef3c6ed6ce99f7934d79874bb705593d0_p.aux.
! Missing $ inserted.
<inserted text> 
                $
l.10 \mu
        _0
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
! Missing $ inserted.
<inserted text> 
                $
l.11 \end{document}
                   
[1] (./latex_d63a06fef3c6ed6ce99f7934d79874bb705593d0_p.aux) )
(see the transcript file for additional information)
Output written on latex_d63a06fef3c6ed6ce99f7934d79874bb705593d0_p.dvi (1 page,
 252 bytes).
Transcript written on latex_d63a06fef3c6ed6ce99f7934d79874bb705593d0_p.log.

and a sample mean

latex error! exitcode was 1 (signal 0), transscript follows:

This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_9f9a03400b0ea551f54495181836fb66e821574a_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
No file latex_9f9a03400b0ea551f54495181836fb66e821574a_p.aux.
)
Runaway argument?
{x \end {document} 
! File ended while scanning use of \mathaccentV.
<inserted text> 
                \par 
<*> ...a03400b0ea551f54495181836fb66e821574a_p.tex
                                                  
! Emergency stop.
<*> ...a03400b0ea551f54495181836fb66e821574a_p.tex
                                                  
No pages of output.
Transcript written on latex_9f9a03400b0ea551f54495181836fb66e821574a_p.log.

}. There is a function z.test() in the BSDA package

• It is necessary that the population variance

latex error! exitcode was 1 (signal 0), transscript follows:

This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_b6cf8a9354a8e0540d0d6c2e0e7d9fea6d962c1f_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
No file latex_b6cf8a9354a8e0540d0d6c2e0e7d9fea6d962c1f_p.aux.
! Missing $ inserted.
<inserted text> 
                $
l.10 \sigma
           ^2
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
! Missing $ inserted.
<inserted text> 
                $
l.11 \end{document}
                   
[1] (./latex_b6cf8a9354a8e0540d0d6c2e0e7d9fea6d962c1f_p.aux) )
(see the transcript file for additional information)
Output written on latex_b6cf8a9354a8e0540d0d6c2e0e7d9fea6d962c1f_p.dvi (1 page,
 256 bytes).
Transcript written on latex_b6cf8a9354a8e0540d0d6c2e0e7d9fea6d962c1f_p.log.

is known.

• The test is accurate if the population is normally distributed. If the population is not normal, the test will still give an approximate guide.

Simulation Exercises

• Now sample 100 values from a Normal distribution with mean 10 and standard deviation 2 and use a z-test to compare it against the population mean 10. What is the p-value?
• Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop! Record at least the p-values and the estimated differences! Use table() to count the p-vals below 0.05. What type of error do you associate with it? What is the smallest absolute difference with a p-value below 0.05?
• Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05?

Simulation Exercises -- Solutions

• Now sample 100 values from a Normal distribution with mean 10 and standard deviation 2 and use a z-test to compare it against the population mean 10. What is the p-value? What the estimated difference?

   1 > ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10)["pval"]
   2 pval 
   3 > ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10)["diff"]
   4 diff 
   5 > ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10)[c("pval","diff")]
   6 pval   diff 

• Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop. Record at least the p-values and the estimated differences! Transform the result into a data frame.

using replicate()\footnotesize

   1 > res <- replicate(1000, ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10))
   2 > res <- as.data.frame(t(res))
   3 > head(res)
   4 diff       Z   pval   n
   5 1 -0.2834 -1.4170 0.1565 100
   6 2  0.2540  1.2698 0.2042 100
   7 3 -0.1915 -0.9576 0.3383 100
   8 4  0.1462  0.7312 0.4646 100
   9 5  0.1122  0.5612 0.5747 100
  10 6 -0.0141 -0.0706 0.9437 100

• Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop. Record at least the p-values and the estimated differences! Transform the result into a data frame.

using replicate() II\footnotesize

   1 > res <- replicate(1000, ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10),
   2 +                        simplify = F)
   3 > res <- as.data.frame(Reduce(rbind,res))
   4 > head(res)
   5 diff       Z   pval   n
   6 init -0.0175 -0.0874 0.9304 100

• Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop. Record at least the p-values and the estimated differences! Transform the result into a data frame.

using for()

   1 > res <- matrix(numeric(2000),ncol=2)
   2 > for(i in seq.int(1000)){
   3 +     res[i,] <- ztest(rnorm(100,mean=10,sd=2),x.sd=2,mu=10)[c("pval","diff")] }
   4 > res <- as.data.frame(res)
   5 > names(res) <- c("pval","diff")
   6 > head(res)
   7 pval    diff
   8 1 0.0591 -0.3775
   9 2 0.2466  0.2317
  10 3 0.6368  0.0944
  11 4 0.5538 -0.1184
  12 5 0.9897 -0.0026
  13 6 0.7748  0.0572

• Use table() to count the p-vals below 0.05. What type of error do you associate with it? What is the smallest absolute difference with a p-value below 0.05?

   1 > table(res$pval < 0.05)
   2 FALSE  TRUE 
   3 960    40 
   4 > tapply(abs(res$diff),res$pval < 0.05,summary)
   5 > min(abs(res$diff[res$pval<0.05]))
   6 [1] 0.3928

• Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05?

   1 > res2 <- replicate(1000, ztest(rnorm(1000,mean=10,sd=2),
   2 +                              x.sd=2,mu=10))
   3 > res2 <- as.data.frame(t(res2))
   4 > head(res2)
   5 diff       Z   pval    n
   6 1 -0.0731 -1.1559 0.2477 1000
   7 2  0.0018  0.0292 0.9767 1000
   8 3  0.0072  0.1144 0.9089 1000
   9 4 -0.1145 -1.8100 0.0703 1000
  10 5 -0.1719 -2.7183 0.0066 1000
  11 6  0.0880  1.3916 0.1640 1000

• Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05?

   1 > table(res2$pval < 0.05)
   2 FALSE  TRUE 
   3 946    54 
   4 > tapply(abs(res2$diff),res$pval < 0.05,summary)

Simulation Exercises Part II

* Concatenate the both resulting data frames from above using rbind() * Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram) * What is the message?

Simulation Exercises -- Solutions

• Concatenate the both resulting data frames from above using rbind()
• Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram)

   1 > res <- rbind(res,res2)  
   2 > require(ggplot2)
   3 > ggplot(res,aes(x=pval)) +
   4 +     geom_histogram(bin=0.1,fill="forestgreen") +
   5 +     facet_grid(~ n)
   6 > ggsave("hist.png")

• Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram)

   1 > ggplot(res,aes(x=diff,colour=factor(n))) +
   2 +     geom_density(size=3)
   3 > ggsave("dens.png")

Simulation Exercises -- Solutions

   1 > set.seed(1)
   2 > x <- rnorm(12)
   3 > t.test(x,mu=0) ## population mean 0
   4 t = 1.1478, df = 11, p-value = 0.2754
   5 alternative hypothesis: true mean is not equal to 0
   6 95 percent confidence interval:
   7 sample estimates:
   8 mean of x 

   1 > t.test(x,mu=1) ## population mean 1
   2 t = -3.125, df = 11, p-value = 0.009664
   3 alternative hypothesis: true mean is not equal to 1
   4 95 percent confidence interval:
   5 sample estimates:
   6 mean of x 

t-tests

A t-test is any statistical hypothesis test in which the test statistic follows a \emph{Student's t distribution} if the null hypothesis is supported.

• \emph{one sample t-test}: test a sample mean against a population mean

One Sample t-test

Two Sample t-tests

There are two ways to perform a two sample t-test in R:

• given two vectors x and y containing the measurement values from the respective groups t.test(x,y)

• given one vector x containing all the measurement values and one vector g containing the group membership #!latex t.test(x \sim g) (read: x dependend on g)

Two Sample t-tests: two vector syntax

   1 > set.seed(1)
   2 > x <- rnorm(12)
   3 > y <- rnorm(12)
   4 > g <- sample(c("A","B"),12,replace = T)
   5 > t.test(x,y)
   6 t = 0.5939, df = 20.012, p-value = 0.5592
   7 alternative hypothesis: true difference in means is not equal to 0
   8 95 percent confidence interval:
   9 sample estimates:
  10 mean of x  mean of y 

Two Sample t-tests: formula syntax

   1 > t.test(x ~ g)
   2 t = -0.6644, df = 6.352, p-value = 0.5298
   3 alternative hypothesis: true difference in means is not equal to 0
   4 95 percent confidence interval:
   5 sample estimates:
   6 mean in group A mean in group B 

Welch/Satterthwaite vs. Student

• if not stated otherwise t.test() will not assume that the variances in the both groups are equal
• if one knows that both populations have the same variance set the var.equal argument to TRUE to perform a student's t-test

Student's t-test

   1 > t.test(x, y, var.equal = T)
   2 t = 0.5939, df = 22, p-value = 0.5586
   3 alternative hypothesis: true difference in means is not equal to 0
   4 95 percent confidence interval:
   5 sample estimates:
   6 mean of x  mean of y 

t-test

• the t-test, especially the Welch test is appropriate whenever the values are normally distributed

• it is also recommended for group sizes #!latex \geq 30 (robust against deviation from normality)

Exercises

* use a t-test to compare TTime according to Stim.Type, visualize it. What is the problem? * now do the same for Subject 1 on pre and post test (use filter() or indexing to get the resp. subsets) * use the following code to do the test on every subset Subject and testid, try to figure what is happening in each step:\tiny

   1 tob <- t.test(x$TTime ~ x$Stim.Type)
   2 tmp <- data.frame(
   3 Subject = unique(x$Subject),
   4 testid = unique(x$testid),
   5 pval = tob$p.value,
   6 alternative = tob$alternative,
   7 res <- Reduce(rbind,tmp.l)

* make plots to visualize the results. * how many tests have a statistically significant result? How many did you expect? Is there a tendency? What could be the next step?

Exercises - Solutions

• use a t-test to compare TTime according to Stim.Type, visualize it. What is the problem?

   1 > t.test(data$TTime ~ data$Stim.Type)
   2 t = -6.3567, df = 9541.891, p-value = 2.156e-10
   3 alternative hypothesis: true difference in means is not equal to 0
   4 95 percent confidence interval:
   5 sample estimates:
   6 mean in group hit mean in group incorrect 
   7 > ggplot(data,aes(x=Stim.Type,y=TTime)) +
   8 +     geom_boxplot()

Exercises - Solutions

• now do the same for Subject 1 on pre and post test (use filter() or indexing to get the resp. subsets)

   1 > t.test(data$TTime[data$Subject==1 & data$testid=="test1"] ~
   2 +        data$Stim.Type[data$Subject==1 & data$testid=="test1"])
   3 t = -0.5846, df = 44.183, p-value = 0.5618
   4 alternative hypothesis: true difference in means is not equal to 0
   5 95 percent confidence interval:
   6 sample estimates:
   7 mean in group hit mean in group incorrect 
   8 > t.test(data$TTime[data$Subject==1 & data$testid=="test2"] ~
   9 +        data$Stim.Type[data$Subject==1 & data$testid=="test2"])
  10 t = -1.7694, df = 47.022, p-value = 0.08332
  11 alternative hypothesis: true difference in means is not equal to 0
  12 95 percent confidence interval:
  13 sample estimates:
  14 mean in group hit mean in group incorrect 

Exercises - Solutions

• make plots to visualize the results

   1 > ggplot(data,aes(x=testid,y=TTime)) +
   2 +    geom_boxplot(aes(fill=Stim.Type)) +
   3 +    facet_wrap(~Subject)
   4 > ggplot(data,aes(x=factor(Subject),y=TTime)) +
   5 +    geom_boxplot(aes(fill=Stim.Type)) +
   6 +    facet_wrap(~testid)

Exercises - Solutions

• how many tests have an statistically significant result? How many did you expect?

   1 > table(res$pval < 0.05)
   2 FALSE  TRUE 
   3 165    22 
   4 > prop.table(table(res$pval < 0.05))
   5 FALSE      TRUE 
   6 > 

Exercises - Solutions

• What could be the next step?

   1 tob <- t.test(x$TTime ~ x$Stim.Type)
   2 tmp <- data.frame(
   3 Subject = unique(x$Subject),
   4 testid = unique(x$testid),
   5 pval = tob$p.value,
   6 alternative = tob$alternative,
   7 res <- Reduce(rbind,tmp.l)