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BMS 617. Lecture 7 – T-tests. T-tests. T-tests refer to a family of statistical tests, in which a mean, or difference in two means, is assumed to be sampled from a T-distribution As a statistical test, T-tests compute p-values

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bms 617

BMS 617

Lecture 7 – T-tests

Marshall University Genomics Core Facility

t tests
T-tests
  • T-tests refer to a family of statistical tests, in which a mean, or difference in two means, is assumed to be sampled from a T-distribution
  • As a statistical test, T-tests compute p-values
    • The probability of seeing a mean, or difference of means, this large, assuming the null hypothesis is true
    • Interpreting a t-test involves knowing the null hypothesis

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types of t test
Types of T-test
  • One-class T-test:
    • Null hypothesis is that the mean of a set of values is equal to some fixed value
  • Two-class T-tests:
    • Two groups of values
      • Unpaired T-test
        • Most common T-test
        • Null hypothesis is that the values in each group are sampled from distributions with equal means
      • Paired T-test
        • Each sample in the first group is paired with a sample in the second group
        • Null hypothesis is that the mean of the difference between each pair is zero

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example one class t test
Example: One-class T-test
  • Recall our body-temperature data from earlier
    • n=130 samples of body temperature
    • Mean m=36.82C, SD s=0.41C
  • We wanted to use this to test the hypothesis that mean body temperature was μ=37C
  • Under the assumptions of the one-class t-test, the value t=(μ-m)/(s/√n) follows a t-distribution with n-1 degrees of freedom
  • For this example, t=5.006

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computing a p value for a given t
Computing a p-value for a given t
  • To get a p-value for t=5.006, we either use software or a table
    • Need to know the degrees of freedom
    • In this case, d.f.=130-1=129
  • Tables either give the probability value for a given t and df, or critical t-values for given probabilities and df
  • From tables or software, the probability that t<=5.006 is p=0.9999991
  • We want to know the probability of seeing a result this extreme, assuming the mean is 37C, i.e. assuming t follows a t-distribution with 129 d.f.
  • This is the probability either that t>5.006 or that t<-5.006
    • A two-tailed, one-class t-test
  • P(t>5.006)=1-0.9999991=0.0000009
  • So P(t<-5.006)=0.0000009
  • and p=2 x 0.0000009 = 0.0000018

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assumptions for a one class t test
Assumptions for a one-class t-test
  • The one-class t-test is accurate under the following assumptions:
    • The samples are random (or representative)
    • The observations are independent
    • The data are accurate
    • The data are sampled from a population that is normally distributed

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one class t test and the confidence interval
One-class T-test and the Confidence Interval
  • We saw earlier how to compute a confidence interval for the mean of this data set
    • Calculate w=t*s/√n
    • The confidence interval is from m-w to m+w
    • t* is the value from the t-distribution for which P(t>t* or t<-t*)=1-confidence
      • e.g. for a 95% confidence interval we want P(t>t* or t<-t*) =0.05
      • So P(t>t*)=0.025. From tables or software, t*=1.979.
      • This gives a 95% confidence interval of [36.75, 36.89]
      • Knowing the 95% confidence interval does not contain the null hypothesis value of 37 is equivalent to knowing the p-value is less than 0.05

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two class unpaired t test example
Two-class unpaired T-test: Example
  • For an example of a two-class, unpaired T-test, consider the GRHL2 expression data we saw earlier from Cieply et al., Cancer Research 2012.
  • Compared expression of GRHL2 in different breast cancer cell lines, classified as Basal A, Basal B, or Luminal.
    • Compare the Basal A expression to the Basal B expression

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grhl2 expression data
GRHL2 Expression Data

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grhl2 expression data1
GRHL2 Expression Data

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how an unpaired t test works
How an unpaired t-test works

An unpaired t-test works by computing the difference of the means of the two samples

Assuming the null hypothesis – that the difference of the two means is zero – the difference of the sample means, divided by a pooled standard error of the mean, will be distributed with a t-distribution

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unpaired t test for the grhl2 data
Unpaired t-test for the GRHL2 data

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assumptions for the unpaired t test
Assumptions for the unpaired T-test
  • The unpaired T-test works along essentially the same assumptions as the one-class T-test:
    • The samples are random or representative
    • The observations are independent
    • The data are accurate
    • The values in the populations are at least approximately normally distributed
  • Additionally, the t-test we used here assumes:
    • The the populations have the same standard deviation

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assumption of equal variances
Assumption of equal variances
  • In the Basal A vs Basal B GRHL2 comparison, the Basal B samples have higher SD (0.7859) than the Basal A samples (0.4463)
  • The t-test we ran assumed the samples came from populations with equal variances (i.e. equals standard deviations)
  • A test can be run to see if the data are consistent with the assumption of equal variances
    • The distribution of the square of the ratio of the standard deviations is known under the assumption that the population variances are equal

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if the assumption of equal variances is violated
If the assumption of equal variances is violated
  • A modified t-test can be used, which doesn’t make the assumption of equal variances
    • Called the “Welch T-test”
    • Has less power than the standard unpaired t-test
  • As usual, testing your data set in order to decide which test to use can give misleading results
    • Typically will give over-optimistic p-values
  • In the ideal world, we would run an experiment specifically to determine if the assumption of equal variances holds, then use that to determine how to analyze our real experiment

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rules of thumb for the assumption of equal variances
Rules of thumb for the assumption of equal variances
  • Unequal variances will only badly affect the t-test if the number of samples in each group is small and unequal
    • In other cases the t-test is very robust to violations of this assumption
  • In practice, I do the following:
    • If the number of samples is equal, I use the regular t-test
    • If the number of samples in both groups is at least 5, no matter if they are equal, I use the regular t-test
    • If there is reason to believe the variances should be equal (e.g. if all the variance comes from technical replicates), I use the standard t-test
    • Otherwise, I use the Welch T-test

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95 confidence intervals and unpaired t tests
95% Confidence Intervals and Unpaired t-tests

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95 confidence intervals and unpaired t tests1
95% Confidence Intervals and unpaired T-tests
  • The unpaired t-test results computed the difference between the means and the 95% confidence interval for that difference
    • For this example the 95% confidence interval of the difference of the means was [-2.373, -1.348]
  • If this confidence interval doesn’t contain zero, this is equivalent to p<0.05
  • We can also compute the confidence interval for each mean independently
  • If these confidence intervals do not overlap, then the p value is definitely less than 0.05
    • In fact, it must be a lot less…
  • If the confidence intervals do overlap, then p may or may not be less than 0.05
    • Cannot deduce anything from the error bars in this case

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other error bars and statistical significance
Other error bars and statistical significance
  • What if the bar chart uses SD or SEM for the error bars
  • SD tells us about the amount of scatter in the data
    • Nothing about the precision with which the mean is measured
  • Overlapping, or non-overlapping, SD error bars have nothing to do with statistical significance
  • SEM measures the precision with which we approximate the mean
    • But interpretation depends on knowing the sample size
    • We can deduce the following:
      • If SEM error bars overlap, then the difference is definitely not statistically significant at p=0.05 (in fact, p is much bigger…)
      • If SEM error bars do not overlap, the p may or may not be less than 0.05

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error bars and statistical significance summary
Error bars and statistical significance summary

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paired t tests
Paired t-tests
  • Paired t-tests are used when the comparison is between samples which are paired between the groups
    • Before and after treatments on a set of patients
      • Pair the “before value” on patient A with the “after value” on patient A
      • The “before value” on patient B with the “after value” on patient B, etc
    • Studies in which subjects are recruited to two groups in a matched fashion
      • Match a control patient with a treatment patient based on age, sex, weight, height…
      • Difficult type of study to perform
    • Twin or sibling studies
    • Lab experiments in which treated and control samples are handled in parallel
      • Plate cells, divide into two, treat one half and use the other as control
      • Repeat the next day with another plate, etc

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example
Example
  • In a recent experiment, we performed expression profiling on a set of eight mice
    • Should not use t-test here without correcting for multiple hypotheses, but this is a good example for demonstration
  • Four litters of mice were bred, and two male mice selected (at random if necessary) from each litter
  • One mouse from each pair was treated and one was used as a control
  • Analyzing these data with a paired test has the potential to eliminate any litter-litter variation

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example data
Example data
  • Actual data are read counts for the gene of interest
    • After sequencing, align all reads to the genome
    • Count the number of reads that align to each gene for each sample

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paired t test
Paired t-test
  • There are two distinct null hypotheses we can make about paired data
    • Both say “the data is no different between the two groups”
    • One is that the difference between the values in the groups is zero (“paired t-test”)
    • The other is that the ratio of the values in the groups is 1 (“ratio paired t-test”)

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paired t test results
Paired T-test results

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how a paired t test works
How a paired t-test works
  • A paired t-test is really just a one-class t-test!
    • Computes the differences for each pair
    • And then tests the null hypothesis that those differences are samples from a normal distribution with mean zero
  • The confidence interval is just the confidence interval of the mean differences

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ratio paired t tests
Ratio paired t-tests
  • The ratio paired t-test tests the null hypothesis that the ratio of the paired values is 1
  • This is done simply by a mathematical trick: take the log of all the ratios
    • Then perform a regular paired t-test with the log ratios instead of the differences
    • Log(1)=0
    • Software performing a ratio paired t-test takes care of computing the logs, performing the test, and then transforming the mean difference of log ratios, confidence interval of that mean, etc, back to ratio values

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graphing paired data
Graphing paired data
  • Plotting bar charts, or even column-scatter plots, of paired data does not show the pairing between the data values
  • A better presentation is a connected column scatter plot
    • Column scatter plot with lines connecting the paired data points

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connected column scatter plot
Connected column scatter plot

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t test summary
t-test summary
  • t-tests are a family of statistical hypothesis tests
  • Generate a p-value
    • Remember how to interpret!
  • Null hypotheses:
    • For a one class t-test, the null hypothesis is that the samples are drawn from a normally-distributed population with a specified mean
    • For an unpaired t-test, the null hypothesis is that the samples are drawn from two normally-distributed populations with equal means
    • For a paired t-test, the null hypothesis is that the differences between matched values are samples of a normally-distributed population with mean zero
      • Equivalent to a one class t-test on the differences between matched values

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