Anja Bråthen Kristoffersen Biomedical Research Group · Famous hypotheses test example: The Design of Experiments (1935), Sir Ronald A. Fisher –A tea party in Cambridge in the

Basic statistical tests in R

Anja Bråthen Kristoffersen

Biomedical Research Group

Outline

• Example: Tea testing

– Hypotese testing, type I and type II error

• More tests and how to find and read help files for different tests in R

•

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Famous hypotheses test example:

The Design of Experiments (1935), Sir Ronald A. Fisher

– A tea party in Cambridge in the 1920ties

– A lady claims that she can taste whether milk is poured in cup before or after the tea

– All professors agree: impossible

– Fisher: this is statistically interesting!

He organized a test

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The lady tasting tea • Test with 8 trials, 2 cups in each trial

– In each trial: guess which cup had the milk poured in first

• Binomial experiment

– Independent trials

– Two possible outcomes, she guesses right cup (success), wrong cup (failure)

– Constant probability of success in each trial

• X = number of correct guesses in 8 trials, each with probability of success p

– X is Binomially (8,p) distributed

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The lady tasting tea cont. • The null (conservative) hypothesis

– The one we initially believe in

• The alternative hypothesis

– The new claim we wish to test

• She has no special ability to taste the difference

• She has a special ability to taste the difference

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𝑝 = 0.5

𝑝 > 0.5

How many right to be convinced We expect maybe 3, 4 or 5 correct guesses if she has no special ability

• Assume 7 correct guesses

– Is there enough evidence to claim that she has a special ability? If 8 correct guesses this would have been even more obvious!

• What if only 6 correct guesses?

– Then it is not so easy to answer YES or NO

• Need a rule that says something about what it takes to be convinced.

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How many right to be convinced?

• Rule: We reject H0 if the observed data have a small probability under H0 (given H0 is true).

• Compute the p-value.

– The probability to obtain the observed value or something more extreme, given that is true

– NB! The p-value is NOT the probability that is true

Small p-value: reject the null hypothesis

Large p-value: keep the null hypothesis 2014.01.15 7

The lady tasting tea, cont. • Say: she identified 6 cups correctly

• P-value – The probability to obtain the observed value or something more

extreme, given that H0 is true 𝑃 𝑋 ≥ 6 𝐻0 𝑡𝑟𝑢𝑒 =

𝑃 𝑋 = 6 𝑝 = 0.5 + 𝑃 𝑋 = 7 𝑝 = 0.5 + 𝑃 𝑋 = 8 𝑝 = 0.5 = 𝑑𝑏𝑖𝑛𝑜𝑚 6, 8, 0.5 + 𝑑𝑏𝑖𝑛𝑜𝑚 7, 8, 0.5 + 𝑑𝑏𝑖𝑛𝑜𝑚 8, 8, 0.5 =

sum(dbinom(6:8, 8, 0.5)) = 0.1445

• Is this enough to be convinced?

• Need a limit. – we must know about the types of errors we can make.

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Two types of error

• Type I error most serious

– Wrongly reject the null hypothesis

– Example:

• person is not guilty

• person is guilty

• To say a person is guilty when he is not is far more serious than to say he is not guilty when he is.

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H0 true H1 true

Accept H0 OK Type II error

Accept H1 Type I error OK

When to reject

• Decide on the hypothesis’ level of significance

– Choose a level of significance α

– This guarantees P(type I error) ≤ α

– Example

• Level of significance at 0.05 gives 5 % probability to reject a true

• Reject H0 if P-value is less than α

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Important parameters in hypothesis testing

• Null hypothesis

• Alternative hypothesis

• Level of significance

Must be decided upon before we know the results of the experiment

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The lady tasting tea, cont.

• Choose 5 % level of significance Conduct the experiment

– Say: she identified 6 cups correctly

– Is this evidence enough?

• P-value

– The probability to obtain the observed value or something more extreme, given that H0 is true

𝑃 𝑋 ≥ 6 𝐻0 𝑡𝑟𝑢𝑒 = 𝑠𝑢𝑚(𝑑𝑏𝑖𝑛𝑜𝑚 6: 8, 8, 0.5)= 0.1445

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The lady tasting tea, cont.

• We obtained a p-value of 0.1443

• The rejection rule says

– Reject H0 if p-value is less than the level of significance α

– Since α = 0.05 we do NOT H0 reject

Small p-value: reject the null hypothesis

Large p-value: keep the null hypothesis 2014.01.15 13

The lady tasting tea, cont.

• In the tea party in Cambridge:

– The lady got every trial correct!

• Comment:

– Why does it taste different?

• Pouring hot tea into cold milk makes the milk curdle, but not so pouring cold milk into hot tea*

2014.01.15 14 *http://binomial.csuhayward.edu/applets/appletNullHyp.html Curdle = å skille seg

Area of rejection

• Reject H0 if p-value ≤ α

• Reject H0 if observed value (x) ≥ critical value (xc)

• P(type I error) = P(reject H0 | H0 true) =

P(X ≥ xc|p = 0.5) – xc= 7 → sum(dbinom(7:8, 8, 0.5)) = 0.035 ≤ 0.05

– xc= 6 → sum(dbinom(6:8, 8, 0.5)) = 0.145 > 0.05

Area of rejection: {x: x ≥ xc} → {x: x ≥ 7}

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Type II error

P(type I error) ≤ α P(type II error) = β

• Want both errors as small as possible

– especially type I.

• β is not explicitly given, depends on H1

• There is one β for each possible value of p under H1

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H0 true H1 true

Accept H0 OK Type II error

Accept H1 Type I error OK

Example: type II error

• P(type II error) = P(not reject H0 | H1 true)

– P = 0.7:

P(not reject H0 | p = 0.7) = 1 - P(reject H0 | p = 0.7) =

1 – P(X ≥ 7 | p = 0.7) = 1 - (1 – P(X < 7 | p = 0.7) =

P(X ≤ 6|p = 0.7) = sum(dbinom(1:6, 8, 0.7)) = 0.745

p = 0.7: H0 will wrongly be accept in 74.5% of the tests

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Power of the test

• The probability that a false H0 is rejected

P(reject H0 | H1 true) = 1 - P(accept H0 | H1 true) = 1 - β

• A test with large power has:

– larger probability to draw the right conclusion

– larger probability to reject a false null hypothesis

then a test with low power.

• α and β is connected:

– Decreasing α will give an increased β which again will decrease the power of the test

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Example: power function

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p <- seq(0.6, 1, 0.01)

antall <- length(p)

beta8 <- rep(NA, antall)

for(i in 1:antall){

beta8[i] <- sum(dbinom(1:6, 8, p[i]))

}

power8 <- 1 - beta8

plot(p, power8, type = "l")

http://www.stanford.edu/~stephsus/ basic-stats-tests.pdf

help(shapiro.test)

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shapiro.test()

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chisq.test()

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Reject the null hypotheses and assume that there are differences between the groups

t.test()

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t.test()

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t.test(, paired = TRUE)

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wilcox.test()

• Same mean? But not normally distributed data.

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wilcox.test(, paired = TRUE)

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help(cor.test)

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Multiple hypothesis testing

• Tests are designed such that it has an expected proportion of incorrectly rejected null hypotheses, most often this level is 5%.

• When many tests are done the probability of rejecting a null hypotheses falsely increase, hence we can correct the probabilities according to how many tests that are done.

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Example 10000 genes • Q: is gene g, g = 1, …, 10 000, differentially expressed?

• Gives 10 000 null hypothesis: 𝐻01, 𝐻0

2, … , 𝐻010000

– 𝐻01: gene 1 not differentially expressed

– …

• Assume: no genes differentially expressed

– 𝐻0𝑔

true for all g

• Significance level α ≤ 0.01

– The probability to incorrectly conclude that one gene is differentially expressed is 0.01. e.g. 0.01 * 10000 = 100

expected wrong rejections of 𝐻0𝑔

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Need to control the risk of false positive Type I error

• Corrected p-value:

– The original p-values do not tell the full story.

– Instead of using the original p-values for decision making, we should use corrected ones.

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Different correction methods

• Bonferroni (1935)

– Just multiply all the p-values by the number of tests

– To conservative

• need very small p-value to reject 𝐻0

• give very little power

• Methods that control the family-wise error rate (FWER).

• Methods that control the false discovery rate (FDR).

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Family-Wise Error Rate (FWER)

• Control type I errors at a level α

– Bonferroni

– Sidak

– Bonferroni-Holm

– Westfall & Young

• Use one of these if you are most afraid of getting stuff on your significant list that should not have been there

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False Discovery Rate (FDR)

• Calculate the expected proportion of type I error among the rejected hypotheses

• Technique that applies to a set of p-values

– Benjamini & Hochberg

– Different newer variants of Benjamini & Hochberg

• Use one of these if you are you most afraid of missing out on interesting stuff

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help(p.adjust)

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False discovery rate (fdr)

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