Hypothesis testing. Want to know something about a population Take a sample from that population Measure the sample What would you expect the sample to

Hypothesis testing

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Hypothesis testing

• Want to know something about a population

• Take a sample from that population• Measure the sample• What would you expect the sample to

look like under the null hypothesis?• Compare the actual sample to this

expectation

population

sample

Y = 2675.4

Hypothesis testing

• Hypotheses are about populations

• Tested with data from samples

• Usually assume that sampling is random

Types of hypotheses

• Null hypothesis - a specific statement about a population parameter made for the purposes of argument

• Alternate hypothesis - includes other possible values for the population parameter besides the value states in the null hypothesis

The null hypothesis is usually the simplest

statement, whereas the alternative hypothesis

is usually the statement of greatest

interest.

A good null hypothesis would be interesting if proven wrong.

A null hypothesis is specific; an alternate hypothesis is not.

Hypothesis testing: exampleCan sheep recognize each other?



The experiment and the results

• Sheep were trained to get a reward near a certain other sheep’s picture

• Then placed in a Y-shaped maze



You must choose…

Stating the hypotheses

H0: Sheep go to each face with equal probability (p = 0.5).

HA: Sheep choose one face over the other (p ≠ 0.5).

Estimating the value

• 16 of 20 is a proportion of p = 0.8

• This is a discrepancy of 0.3 from the proportion proposed by the null hypothesis, p =0.5

Null distribution

• The null distribution is the sampling distribution of outcomes for a test statistic under the assumption that the null hypothesis is true

Proportion of correct choices

Probability

0 0.000001

1 0.00002

2 0.00018

3 0.0011

4 0.0046

5 0.015

6 0.037

7 0.074

8 0.12

9 0.16

10 0.18

11 0.16

12 0.12

13 0.074

14 0.037

15 0.015

16 0.0046

17 0.0011

18 0.00018

19 0.00002

20 0.000001

• Test statistic = a quantity calculated from the data that is used to evaluate how compatable the data are with the expectation under the null hypothesis

The null distribution of p

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of correct choices

Frequency

Test statistic = 16

The null distribution of p

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Frequency

Values at least as extreme as the test statistic

• P-value - the probability of obtaining the data* if the null hypothesis were true

*as great or greater difference from the null hypothesis

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Frequency

P = 0.012

P-value

P-value calculation

P

=2*(Pr[16]+Pr[17]+Pr[18]+Pr[19]+Pr[20])

=2*(0.005+0.001+0.0002+0.00002+0.000001)

= 0.012

How to find P-values

• Get test statistic

• Compare with null distribution from:– Simulation– Parametric tests– Non-parametric tests– Re-sampling

Statistical significance

The significance level, α, i s a probability used as a criterion

for rejecti ng the nul l hypothesi .s If th e P-valu e for a test is

less than or equal t o α, the n the null hypothesi s is rejecte .d

α is often 0.05

Significance for the sheep example

• P = 0.012

• P < α, so we can reject the null hypothesis

Larger samples give more information

• A larger sample will tend to give and estimate with a smaller confidence interval

• A larger sample will give more power to reject a false null hypothesis

Hypothesis testing: another exampleThe genetics of symmetry in flowers

Heteranthera - Mud plantain



Stigma and anthers are asymmetric in different genotypes

Can the pattern of inheritance be explained by a single locus

with simple dominance? Model predicts a 3:1 ratio of right-handed

flowers

H0: Right- and left-handed offspring occur at a 3:1 ratio (the proportion of right-handed individuals in the offspring population is p = 3/4)

HA: Right- and left-handed offspring do not occur at a 3:1 ratio (p ≠ 3/4)

Data

Of 27 offspring, 21 were “right-handed” and 6 were “left-handed.”

Estimating the proportion

€

ˆ p =21

27= 0.778

* The “hat” notation denotes an estimate for apopulation parameter from a sample

Number of right-handed flowers in a random sampleof 27

Probability

0-8 09 0.00000510 0.00003111 0.00012412 0.00049213 0.00175214 0.00530115 0.01383016 0.03109417 0.06067318 0.10089119 0.14305120 0.17233921 0.17178222 0.14103423 0.09147724 0.04549925 0.01640926 0.00375927 0.000457

Sampling distribution of null hypothesis

P = 0.83.

The P-value:





Rock-paper-scissors battle

Jargon

Significance level

• A probability used as a criterion for rejecting the null hypothesis

• Called α• If p < α, reject the null hypothesis

• For most purposes, α = 0.05 is acceptable

Type I error

• Rejecting a true null hypothesis

• Probability of Type I error is α (the significance level)

Type II error

• Not rejecting a false null hypothesis

• The probability of a Type II error is

• The smaller , the more power a test has

Power

• The probability that a random sample of a particular size will lead to rejection of a false null hypothesis

• Power = 1-

Reality

Result

Ho true Ho false

Reject Ho

Do not reject Ho correct

correctType I error

Type II error

One- and two-tailed tests

• Most tests are two-tailed tests

• This means that a deviation in either direction would reject the null hypothesis

• Normally α is divided into α/2 on one side and α/2 on the other

Test statistic

One-sided tests

• Also called one-tailed tests

• Only used when one side of the null distribution is nonsensical

• For example, comparing grades on a multiple choice test to that expected by random guessing

Critical value

• The value of a test statistic beyond which the null hypothesis can be rejected

“Statistically significant”

• P < α

• We can “reject the null hypothesis”

We never “accept the null hypothesis”

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Hypothesis testing. Want to know something about a population Take a sample from that population Measure the sample What would you expect the sample to