Introductory Statistics for Laboratorians dealing with High

Throughput Data sets

Centers for Disease Control

Statistics and Order

• Random vs. Accidental• Snowflakes

• Quincunx’ – http://www.stattucino.com/berrie/dsl/Galton.html– http://www.jcu.edu/math/ISEP/Quincunx/Quincunx.html

• Statistical Determinism

Hypothesis Testing

• Statistical methods are used to test scientific hypotheses.

• You already understand this logic, you do this every day.

• Statistical methods simply provide a way to put numbers to your logic.

• That is, compute the chances that you are wrong or right.

Harry and Sue

• This is the story of Harry Heartthrob and his Girlfriend Sue Sweetheart. Harry would like to know that everything is fine between him and Sue.

• Begin by stating two hypotheses: – H0: Things are fine in Harry’s love life.

– Ha: Harry has romantic problems.

• We assume that H0: is true – that things are fine. • This leads us to expect to see certain things when

we observe nature/reality.

Hypotheses

• H0: is the “Null Hypotheis.” It states that nothing is going on.

• Ha: is the “Alternative Hypothesis.” It states that something is going on.

• The null and alternative hypotheses must be mutually exclusive and exhaustive.

• That is: they can’t both be true and they can’t both be false.

Harry and Sue

• Believing that H0: is true (as we assume) leads to certain expectations about what will be observed in reality.

• Specifically Harry will expect there to be “no signs of men” in Sue’s apartment.

• If Harry observes things in her apartment that differ from his expectations he will begin to doubt the truth of H0:

H0: Assume All is OK

Expect no Signs of Men in Her Apartment

Harry’s Love Life

Ha: Harry has Romantic Problems

Proof

• Can you ever prove that H0: is true? NO!!! • What is the strongest evidence that H0: is true?– No indication of men in the apartment!

• Could this happen if H0: is false? YES!! • The strongest evidence for H0: is still weak• We “assume” H0: is true if the evidence it is

false is weak.

Proof

• Can Harry prove that H0: is false? YES!!!• There can be things that are so improbable if H0:

is true that when you observe those things, (XXX) for example, you know H0: is false.

• There can be strong evidence that H0: is false.

• When we see that strong evidence we reject H0:

• Then we say “we conclude Ha: is true” or “we have proved Ha: “

H0: Assume All is OK

Expect no Signs of Men in Her Apartment

Statistically Speaking

Ha: Harry has Romantic Problems

98%

20%

2%

.05%

Reject H0: if the probability of the observed event is small enough if H0: is assumed to be true

H0: Assume All is OK

Expect no Signs of Men in Her Apartment

Statistical Error

Ha: Harry has Romantic Problems

98%

20%

2%

.05%

Type I Error: Reject H0: when it is true

Type II Error: Continue to believe H0: when it is false

Two Types of Error

Type I: too jealous

Type II: too trusting

Statistical Error

Truth about Sue Sweetheart

H0 TrueShe is actually not cheating

H0 False She is cheating her A off.

Harry’s Decision Basedon Observed Data

Reject H0 : Concludes she is Cheating

Type IError (Alpha)Too jealous

CorrectDecision

Fail ToReject H0 : Concludes everything is OK

CorrectDecision

Type IIError (Beta)Too trusting

Quality ControlHave the samples been watered down?

• There is a severe shortage of flu vaccine in the USA this season. However, Canada has a large surplus and they are willing to sell it to us.

• We are a little paranoid and wonder if the reason they have extra is because they watered it down.

• We make a surprise visit to their warehouse and request a sample of the vaccine for evaluation purposes before we commit to purchase the lot.

• They allow us to select 70 vials at random from the whole lot for testing.

Quality Control Example

• All flu vaccine is made to standard specifications. It is all supposed to be 16 m/dl with a standard deviation of 0.4. (That’s what it’s supposed to be if it is not watered down).

• We measure the 70 vials from Canada and get a mean of 15.8 m/dl.

• Is the Canadian surplus watered down?

Step 1: State H0 and Ha

• H0: This sample of 70 vials (with a mean of 15.8) comes from a population with a mean of 16 and a standard deviation of 0.4.– (Everything is fine.)

• Ha: This sample of 70 vials could not have been drawn from a population with a mean of 16 and a standard deviation of 0.4. – (There is a problem.)

Step 2: Select a Region of Rejection

• If the probability of the null hypothesis being true is less than 5 chances in 100 (.05) we will reject it.

• Alpha = .05

Step 3: Make Observations

• Conduct the experiment – make surprise trip to the Canadian warehouse, select vials at random, test each vial.

• Compute the mean for the 70 vials. – mean = 15.8.

Step 4: Test the Null Hypothesis

• What does the Central Limit Theorem tells us about the distribution of means of samples of size N = 70 from a population with a mean of 16 and standard deviation of 0.4.

• Central Limit Theorem says: – Mean of the means of all possible samples should

be 16– Standard Error (Standard Deviation) of the means

is 0.4/sqrt(70) = .048

Step 4: Test the Null Hypothesis

• Use http://davidmlane.com/hyperstat/z_table.html to compute the probability that a mean would be 15.8 or greater if the Sampling Distribution of the Mean has a mean of 16 and a standard deviation of .048.

• 15.8 is 4.17 standard errors (standard deviations) below the mean of the population (16).

• Z = -4.17

Step 4: Test the Null Hypothesis

• The probability of the mean of a sample of size 70 being 15.8 or less is .000015 (15 chances in 1,000,000).

• This is in the region of rejection

• Reject H0 -- There is a problem. This stuff has been watered down.

Region of Rejection for a Sample of size N = 70 from a Population with mean 16 and standard deviation of 0.4

• The region of rejection is anything below 15.921.

• 15.921 cuts of .05 of the distribution.

• There are only 5 chances in 100 of a mean being less than 15.921

• Our mean was 15.8

Types of ErrorTruth about Population from which sample came

H0 True H0 False

Decision Basedon Sample

Reject H0 Type IError (Alpha)

CorrectDecision

Fail ToReject H0

CorrectDecision

Type IIError (Beta)

Error in Diagnostic TestingTruth about person

H0 True Really Don’t Have Disease

H0 False Really Do Have Disease

What Diagnostic Test Tells Us

Positive (Says they have disease)

Type I Error (Alpha)False Positive

True Positive

Negative (Says they don’t have disease)

True NegativeType II Error (Beta)False Negative

FNTPTPySensitivit

FPTN

TNySpecificit