the t-test (happy happy joy!)

Summary from last week

Introducing the free experiment (week 10)

Parametric statistics T-test

Exercises The t-test in SPSS Group problem solving on t-tests

Changes to the course plan – new course book uploded Changes to provide more time for parametric statistics

Compendium is in the press – should be available early next week from the book store at KUA

Emilie guest teacher next 2 weeks while I am on paternity leave

For week 12, regression and correlation, 100+ pages in compendium: No need to read all of it – read the introductions to each chapter, get the feel for the first simple examples – multiple regression and –correlation is for future reference

The purpose of this exercise is to give you the chance to design your own experiment, run it, analyze the results and write a report about your experience

You can experiment with anything! Pick something you care about, something you wonder about: What is the best way to make popcorn? Why do my plants die? Is chat program X faster than program Y? How long should rise boil for maximum tenderness? Does my daughter have a favorite toy? Which beer tastes the best? Is iron or aluminium pans best for boiling water? What is the ideal speed for walking with a coffee-cup? Etc. etc.

Example: Designing a 2*3 factorial experiment to determine the effect of three variables on the amount of popcorn produced Variables: Brand of popcorn (Netto, Irma), size of

batch (100 g, 200g), popcorn-to-oil-ratio (low, high)

Looking into e.g. if more expensive popcorn are worth the price in terms of produced amount?

What combination of variables produces the best result in terms of volume?

Having picked your topic, sit down and define your hypothesis

Consider: IV and DV? How many levels to the IV? What is the expected causal relationship? Directional or non-directional hypothesis? Argument

to support this?

Then design an experiment to test your hypothesis

Choose something that is easy to re-test – you may experience that the test design needs to be redesigned

Design experiments in your groups - on the website is a list of 101 examples of student experiment projects as inspiration

You have 4 weeks to complete the experiment and write a report about what was done and what was learned, problems encountered, how to improve the experiment, etc.

Experiments can be the ”home type” or the ”laboratory type”

As you progress and encounter questions, raise them in the class where they can be discussed, or contact me/Emilie

The only formal requirement is that you MUST use statistics in your analysis! This should be a piece of cake – just identify

the type of experiment you do, then use the appropriate statistical test ...

Similar to the Mousepad experiment – but with more statistics as appropriate E.g.: 2 groups with 2 levels of IV? Use t-test Etc.

Week 10: Prepare topic Week 11+12: Run experiment Week 13: Prepare report and presentation of

experiment Week 14: Present experiment and results (5 minutes)

There is no time set aside to this work in the exercise hours (there may be some time in exercises but do not plan for it)

No page limit on the report – you are expected to use the standard report template and add content as appropriate. Use the textbook + Mousepad experiment reports as a guide.

1. Sample means vary, and hence differences between sample means and population mean varies.

2. Small differences are likely to occur by chance, large differences are not (but can occasionally do so).

3. Small difference -> retain null hypothesis (difference has occurred by chance). Large difference –> reject null hypothesis in favour of experimental hypothesis (difference has not occurred by chance).

4. "Large" is a difference that is likely to occur by chance only 5% of the time or less (p < .05) - a compromise between Type 1 and Type 2 errors.

5. Directional hypothesis versus non-directional hypothesis.

Different types of statistics

Descriptive statistics: Describing a single sample and the population it came from

Inferential statistics: To answer research questions – inference about the world

Parametric statistics = inferential statistical testing methods

Parametric statistics work on the mean -> All data must be interval or ratio level data

Parametric tests also make assumptions about the variance between groups or conditions

For independent-measures (between groups), we assume that variance in one condition is the same as the other: Homogeneity of variance

The spread of scores in each sample should be roughly similar Tested using Levene´s test

For repeated-measures (within subjects), we operate with the sphericity assumption,

Tested using Mauchly´s test Basically the same thing: homogeneity of variance

We also assume our data come from a population with a normal distribution

We can test how much a distribution is similar to the normal distribution using the Kolmogorov-Smirnov test (the vodka test) and the Shapiro-Wilk tests The tests compare the set of scores in the sample to a

normally distributed set of scores with the same mean and standard deviation

If the test is non-significant (p>0.05) the distribution of the sample is NOT significantly different from a normal distribution (i.e. it is normal)

If p<0.05, the distribution of the sample is significantly different from normal (e.g. positively or negatively skewed).

We can run Kolmogorov-Smirnov and Shapiro-Wilk tests in SPSS

The most important is the Kolmogorov-Smirnov Test (K-S-test)

SPSS produces an output that includes the test statistic itself (D), the degrees of freedom (df) (= the sample size) and the significance value of the test (sig.).

If the significance of the K-S-test is less than .05, the distribution deviates significantly from the normal

We have run an experiment with two groups (e.g. control and experiment groups)

We have sample data, and we can use descriptive statistics to calcuate the means, SDs etc. etc.

But how do we find out if the two samples are significantly different? I.e.: If our experiment was a success? Our manipulation of IV caused a variation in DV

larger than the random variance

The simplest experimental design is to have two conditions: an "experimental“ condition in which subjects receive some kind of treatment, and a "control" condition in which they do not.

We want to compare performance in the two conditions.

We use a t-test to help us to decide whether the difference between the conditions is "real" or whether it is due merely to chance fluctuations

The t-test enables us to decide whether the mean of one condition is really different from the mean of another condition

We use the t-test in the simplest experimental condition: 2 groups to compare Sample-sample (or sample-population)

The test statistic is called ”t” – it has its own frequency distribution which varies with sample size

There are two types of t-test Independent t-test: 2 groups with different participants

[independent measures design/between-groups] Dependent t-test: 2 groups with same participants

[repeated measures design/within-subject]

In both cases, we have one independent variable The thing we manipulate in our

experiment), with two levels (the two different conditions of our experiment). ▪ Small mouse pad or big mouse pad

We have one dependent variable The thing we actually measure).

▪ Task completion time in seconds

1) Differences between extraverts and introverts in performance on a memory test. The independent variable (I.V.) is "personality type", with

two levels - introversion and extraversion - and the dependent variable (D.V.) is the memory test score

An independent t-test would be appropriate here

2) The effects of alcohol on reaction-time performance. The I.V. is "alcohol consumption", with two levels - drunk

and sober - and the D.V. is reaction-time performance A dependent t-test could be used here; each subject's

reaction time could be measured twice, once while they were drunk and once while they were sober

There are some considerations underlying the t-test which we need to be aware off to avoid using the test blindly

Understanding how statistical tests operate is important – we need to know how tests operate in order to use them correctly

Rationale of the t-test:

1) We have two sample means – they differ to some extent

Given two sample means, - we want to find out if the sample means come from two populations with the same mean (same population), or from two populations with different means.

2) If null hypothesis, means are identical, if experimental hypothesis, means are significantly dissimilar

X 1 & X 2

population mean

mean of population 1

mean of population 2

Interpretation under the null hypothesis: Samples come from the same population:

Interpretation under the experimental hypothesis: Samples come from different populations:

mean of sample 1

mean of sample 2

mean of sample 1

mean of sample 2

3) in the t-test, we compare the differences we have obtained with the difference we would expect (we assume no difference, null hypothesis)

If we find a big difference between the means, we have either 1) atypical samples [by random chance, we got two

dissimilar samples] 2) the samples are from different populations because their

means are different [our experiment had an effect]

The bigger the difference in sample means, the bigger the chance of the null hypothesis being rejected

4) Because samples can be different by random chance, we cannot just work with the difference of the means

We need some way of calculating the odds of two samples being dissimilar by random chance

We can then “compare” our sample means difference with the chance of this difference occurring

I.e., we need to know the frequency distribution of sample mean differences

For example, say the difference in our two sample means is “243”, we need to know how likely this difference size is in our population?

The frequency distribution of the sample means difference can tell us how likely it is that “243” is the difference between two sample means – e.g. “X%”

If the chance of the difference occurring is small, there is a good chance the difference in sample means is significant.

Recall: Sample means from a population will be normally distributed:

-> higher chance of sample means being similar than not

However sometimes samples do not have similar means:

-> large difference in sample means by chance alone we need to account for this when figuring out if samples

are different!

= 10

M = 8M = 10

M = 9

M = 11

M = 12M = 11

M = 9

M = 10

M = 10

Sample Mean

6 7 8 9 10 11 12 13 14

Fre

quen

cy0

1

2

3

4

Mean = 10SD = 1.22

Sampling distribution of differences between means: A new type of distribution

Population I Population II

X 2

X 1. . . . . . . . . . . .

X 1

X 1

X 2

X 2

μ1 μ2

frequencyof D

X 1

X 2values of −

X 1 X 2 D Note: we want to figure out if Pop. 1 and 2 are the same

I.e. We take all possible sample means and subtract all possible sample means, and map the distribution

The distribution is of course normally distributed

The SD of this distribution = SE of differences [SE because we are dealing with the distribution of sample means – we call it SD when we have just one sample]

Sample Mean

6 7 8 9 10 11 12 13 14F

requ

ency

0

1

2

3

4

Mean = 10SD = 1.22

-> small SE means most pairs of samples from a population will generally have similar means (difference between sample means is small)

-> large SE means that sample means can deviate a lot from population mean, and differences between pairs of samples can be large by chance alone

The SE of the sample means difference frequency distribution gives us an estimate of the extent to which we would expect sample means to be different by chance alone A measure of unsystematic variance in our experiment

T-test is simply difference between means as a function of the degree to which those means would differ by chance alone

Note: If large differences are COMMON in the means of samples from a population, because the normal distribution of sample means is flat, the difference between the samples need to be correspondingly larger to be significant

t =

Observed difference of Sample means

Difference between means undernull hypothesis

-

Estimate of the standard error (SE) of the difference betweenthe two sample means (the unsystematic variance

Recall: Two types of t-test

Independent t-test: 2 groups with different participants [independent measures design/between-groups]

Dependent t-test: 2 groups with same participants [repeated measures design/within-subject]

The dependent t-test is used when the same participants are used in both experimental conditions

POPULATION

N subjects

Level 1 of independent variable A administered

subjectsmeasured ondependentvariable B

Compute

Test your hypothesisH0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0

Repeated measures experiment. To examine the effect of variable A on variable B

N subjects are selected fromthe population

Statistics are computed and hypothesistest carried out to decide if the differencebetween and is due to sampling variabilityor effect of A on B.

X 1

X 2

Level 2 of independent variable A administered

subjectsmeasured ondependentvariable B

D

The subjects are first given Level 1 of theindependent variable A

The same subjects are then given Level 2of the independent variable A

Subjects are measured on dependent variable B.( and s1 are computed from these data)

X 1

Subjects are measured on dependent variable B.( and s2 are computed from these data)

X 2

SD

Experiment on the effects of alcohol on task performance (time in seconds).

Measure time taken to perform the task for subjects when drunk, and when (same subjects are) sober.

Null hypothesis: Alcohol has no effect on time taken: Variation between the drunk sample mean and the sober sample mean is due to sampling variation alone.

i.e. The drunk and sober performance times are samples from the same population.

Quick reminder: Sampling distribution of differences between means

Population level 1 of Awith Alcohol

X 2

X 1. . . . . . . . . . . .

X 1

X 1

X 2

X 2

μ1 μ2

frequencyof D

X 1

X 2values of −

X 1 X 2 D

Population level 2 of Awithout Alcohol

μD

Condition A

Level 1

Condition A

Level 2

Participant With Alcohol

WithoutAlcohol

1 12.4 10.0

2 15.5 14.2

3 17.9 18.0

4 9.7 10.1

5 19.6 14.2

6 16.5 12.1

7 15.1 15.1

8 16.3 12.4

9 13.3 12.7

10 11.6 13.1

Times (in seconds) of participants to complete a motor coordination task

the mean difference between scores in our two samples (should be close to zero if there is no difference between the two conditions)

the predicted average difference between scores in our two samples (usually zero, since we assume the two samples do not differ )

estimated standard error of the mean difference (a measure of how much the mean difference might vary from one occasion to the next randomly).

t(observed) D D (hypothesized)

SD

Condition A

Level 1

Condition ALevel 2

Participant With Alcohol

WithoutAlcohol

Diff. (D)

1 12.4 10.0 2.4

2 15.5 14.2 1.3

3 17.9 18.0 -0.1

4 9.7 10.1 -0.4

5 19.6 14.2 5.4

6 16.5 12.1 4.4

7 15.1 15.1 0.0

8 16.3 12.4 3.9

9 13.3 12.7 0.6

10 11.6 13.1 -1.5

16.0

D

If independent t-test,(2 groups of differentsubjects), we just subtract sample mean 1 from sample mean 2

1. Add up the differences:

2. Find the mean difference:

D 16

D D

N

16

101.6

3a. Estimate of the population standard deviation We need this to calculate the standard error of the mean differences

SD ( D D )2

N 1

SD

SD

N

Standard deviation

Standard errorof sample meansdifferences

Condition A

Level 1

Condition ALevel 2

Participant With Alcohol

WithoutAlcohol

Diff. (D)

1 12.4 10.0 2.4 0.8 0.64

2 15.5 14.2 1.3 -0.3 0.09

3 17.9 18.0 -0.1 -1.7 2.89

4 9.7 10.1 -0.4 -2.0 4.0

5 19.6 14.2 5.4 3.8 14.44

6 16.5 12.1 4.4 2.8 7.84

7 15.1 15.1 0.0 -1.6 2.56

8 16.3 12.4 3.9 2.3 5.29

9 13.3 12.7 0.6 -1.0 1.0

10 11.6 13.1 -1.5 -3.1 9.61

16.0 48.36

D D

(D D )2

D

(D D )2

D 16

101.6

Breaking this calculation downIn steps:

3b. Estimate of the population standard deviation

SD ( D D )2

N 1

48.36

92.318

4. Estimate of the population standard error (the SE of the population of differences between means of samples)

Recall: The SE is the SD of sample means(here it is the standard error of the differences between two sample means – our difference frequency distribution):

SD

SD

N

2.318

100.733

5. Hypothesised difference between the sample means Our null hypothesis is usually that there is no difference between the two sample means. (In statistical terms, that they have come from two identical populations):

D (hypothesised) = 0

6. Work out t:

7. "Degrees of freedom" (df) are the number of subjects minus one: df = N - 1 = 10 - 1 = 9

t(observed) 1.6 0

0.7332.183

8. Find t-critical value of t from a table (at the back of statistics books;also on the course website).

(a) “Two-tailed test”: If we are predicting a difference between Level 1 and 2; find the critical value of t for a "two-tailed" test. With df = 9, critical value = 2.26.

(b) “One-tailed test”: If we are predicting that Level 1 is bigger than 2, (or 1 is smaller than 2), find the critical value of t for a "one-tailed" test. For df = 9, critical value = 1.83.

TWO-Tailed: t-observed (2.183) issmaller than t-critical (2.26)

“There is no significant difference between the times taken to complete the task with or without alcohol”

t(9) = 2.183, p > 0.05

ONE-Tailed: t-observed (2.183) islarger than t-critical (1.83)

“The times taken to complete the task is significantly longer with alcohol than without”

t(9) = 2.183, p < 0.05

8. Find t-critical value of t from a table (at the back of statistics books;also on the course website).

0.025 0.025

uppert-critical value

2.26

lowert-critical value

-2.26

two-taileddf = 9

one-taileddf = 9 0.05

t-critical value1.83

t-observed (2.183)

Using SPSS to do a dependent t-test

DataEntry

Running SPSS (repeated measures t-test)

Running SPSS (repeated measures t-test)

Running SPSS (repeated measures t-test)

Interpreting SPSS output (repeated measures t-test)

Sample meansSample sizes

Correlation strength (effect size) + significance of correlation

Mean difference of samples

SD of the difference between the means

SE of differences between scores

Test results

Degrees of freedom is sample size minus 1 in repeated measures (here: 10 – 1 = 9)

SPSS uses the dg. to calculate the odds that the t-value could occur by chance

POPULATION

Random factorsdetermine whichgroup

N subjects

Group 1:n1 subject

Group 2:n2 subject

Level 1of independentvariable Agiven to allsubjects (n1)

Level 2of independentvariable Agiven to allsubjects (n2)

Group 1subjectsmeasured ondependentvariable B

Group 2subjectsmeasured ondependentvariable B

Compute

Error term1

X 1

X 2Compute

Error term2

Test your hypothesisH0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0

Independent measures experiment. To examine the effect of variable A on variable B

N subjects are selected fromthe population and split into twogroups n1 and n2

n1 + n2 = N

Subjects in each groupreceive identical treatmentexcept different levels of

independent variable A aregiven to each group

Subjects in each groupare measured in the same way

on the dependent variable B

Statistics are computed and hypothesistest carried out to decide if the differencebetween and is due to sampling variabilityor effect of A on B.

X 1

X 2

Experiment on the effects of alcohol on task performance (time in seconds).

Measure time taken to perform the task for one set of subjects when drunk, and a different set of subjects when sober.

Null hypothesis: Alcohol has no effect on time taken: variation between the drunk sample mean and the sober sample mean is due to sampling variation.

i.e. The drunk and sober performance times are samples from the same population.

Drunk X1 Sober X2

Participant 1 13.0 Participant 1 11.1

Participant 2 16.5 Participant 2 13.5

Participant 3 16.9 Participant 3 11.0

Participant 4 19.7 Participant 4 9.1

Participant 5 17.6 Participant 5 13.3

Participant 6 17.5 Participant 6 11.7

Participant 7 18.1 Participant 7 14.3

Participant 8 17.3 Participant 8 10.8

Participant 9 14.5 Participant 9 12.6

Participant 10 13.3 Participant 10 11.2

Subject group 1 Subject group 2

the difference between samples means (should be close to zero if there is no difference between the two conditions)

the predicted average difference between scores in our two samples (usually zero, since we assume the two samples don’t differ )

estimated standard error of the difference between the means (a measure of how much the difference between means might vary from one occasion to the next).

t(observed) (X 1 X 2) (1 2)hypothesized

estimatedX 1 X 2

Using SPSS to do an independent t-test

DataEntry

Note 2 samples

Running SPSS (independent measures t-test)

Running SPSS (independent measures t-test)

Running SPSS (independent measures t-test)

Running SPSS (independent measures t-test)

Defining who belongs to which sample

SPSS output (independent measures t-test)

t is calculated by dividing difference in means with standard error: 4.58/0.84359

Row 1 left show result of Levene´s test – tests the hypothesis that variance in the twosamples is equal. If Levene´s test is significant at p<0.05 the assumption of homogenity of variance in the samples has been violated (this is annoying). If not, we assume equal variance (use row 1)

Sig. is < than .05, so there is a significant difference between alcohol/no alcohol on performance

Degrees of freedom is the sum of the sample sizes minus the number of samples (here 10+10-2 = 18)

SPSS uses the dg. To find out which t-distribution to use, and thus to calculate the odds that the t-value could occur by chance t-distributions vary by sample size

We use the t-test when, given two sample means,

, we want to find out if the sample means come from two populations with the same mean, or from two populations with different means.

t-statistic = the ratio of the difference between means divided by an estimate of the standard error of the frequency distribution of the difference of means

t-test is simply the difference between the means of two samples/groups as a function of the degree to which those means would differ by chance alone The t-test informs us if the sample means of two conditions is

large enough not to be a chance result

X 1 & X 2

1) When to use a t-test2) What t-test to use (dependent,

independent)3) That the t-test compares

difference of sample means, as a function of this difference occuring by random chance

4) How to use SPSS to run a t-test and interpret the result

Calculating effect sizes

Is the effect we have found substantive or small?

We can convert t-statistic into an effect size [”r”]

For our independent t-test example (t=5.429; dg.=18)

See ya in 2 weeks!