statistics (cont.) psych 231: research methods in psychology

Statistics (cont.)

Psych 231: Research Methods in Psychology

Announcements

Journal Summary 2 assignment Due in class this week (Wednesday, Nov. 13th)

Group projects Plan to have your analyses done before Thanksgiving break,

GAs will be available during lab times to help Poster sessions are last lab sections of the semester (last

week of classes), so start thinking about your posters. I will lecture about poster presentations on Monday (a week from today).

Samples and Populations

2 General kinds of Statistics Descriptive statistics

• Used to describe, simplify, & organize data sets

• Describing distributions of scores

Inferential statistics• Used to test claims about the

population, based on data gathered from samples

• Takes sampling error into account. Are the results above and beyond what you’d expect by random chance?

Sample

Inferential statistics used to generalize back

Population

Statistics

2 General kinds of Statistics Descriptive statistics

• Used to describe, simplify, & organize data sets

• Describing distributions of scores

Inferential statistics• Used to test claims about the

population, based on data gathered from samples

• Takes sampling error into account. Are the results above and beyond what you’d expect by random chance?

Sample

Inferential statistics used to generalize back

Population

Inferential Statistics

Purpose: To make claims about populations based on data collected from samples What’s the big deal?

Example Experiment: Group A - gets treatment to improve memory Group B - gets no treatment (control)

After treatment period test both groups for memory

Results: Group A’s average memory score is 80% Group B’s is 76%

Is the 4% difference a “real” difference (statistically significant) or is it just sampling error?


After treatment period test both groups for memory

Results: Group A’s average memory score is 80% Group B’s is 76%


Population

Sample ATreatmentX = 80%

Sample BNo TreatmentX = 76%

Testing Hypotheses

Step 2: Set your decision criteria Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis

“Reject H0”

“Fail to reject H0”

Step 1: State your hypotheses

Testing Hypotheses


• “There are no differences (effects)”

• Generally, “not all groups are equal”

You aren’t out to prove the alternative hypothesis (although it feels like this is what you want to do)

If you reject the null hypothesis, then you’re left with support for the alternative(s) (NOT proof!)

This is the hypothesis that you are testing

This is the hypothesis that you are testing

Null hypothesis (H0)

Alternative hypothesis(ses) (HA)

Testing Hypotheses


In our memory example experiment Null H0: mean of Group A = mean of Group B Alternative HA: mean of Group A ≠ mean of Group B

(Or more precisely: Group A > Group B)

It seems like our theory is that the treatment should improve memory.

That’s the alternative hypothesis. That’s NOT the one the we’ll test with inferential statistics.

Instead, we test the H0

In our memory example experiment Null H0: mean of Group A = mean of Group B Alternative HA: mean of Group A ≠ mean of Group B

(Or more precisely: Group A > Group B)

It seems like our theory is that the treatment should improve memory.

That’s the alternative hypothesis. That’s NOT the one the we’ll test with inferential statistics.

Instead, we test the H0

Testing Hypotheses

Step 2: Set your decision criteria Your alpha level will be your guide for when to:

• “reject the null hypothesis”• “fail to reject the null hypothesis”


This could be correct conclusion or the incorrect conclusion• Two different ways to go wrong

• Type I error: saying that there is a difference when there really isn’t one (probability of making this error is “alpha level”)

• Type II error: saying that there is not a difference when there really is one

Error types

Real world (‘truth’)

H0 is correct

H0 is wrong

Experimenter’s conclusions

Reject H0

Fail to Reject H0

Type I error

Type II error

Error types: Courtroom analogy


Defendant is innocent

Jury’s decision

Find guilty

Type I error

Type II error

Defendant is guilty

Find not guilty

Error types

Type I error (α): concluding that there is an effect (a difference between groups) when there really isn’t. Sometimes called “significance level” We try to minimize this (keep it low) Pick a low level of alpha Psychology: 0.05 and 0.01 most common

Type II error (β): concluding that there isn’t an effect, when there really is. How likely are you able to detect a difference if it is there Related to the Statistical Power of a test

Testing Hypotheses

Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics

Descriptive statistics (means, standard deviations, etc.) Inferential statistics (t-tests, ANOVAs, etc.)

Step 5: Make a decision about your null hypothesis Reject H0 “statistically significant differences” Fail to reject H0 “not statistically significant differences”

Step 1: State your hypotheses Step 2: Set your decision criteria

Statistical significance

“Statistically significant differences” When you “reject your null hypothesis”

• Essentially this means that the observed difference is above what you’d expect by chance

• “Chance” is determined by estimating how much sampling error there is

• Factors affecting “chance”• Sample size

• Population variability

Sampling error

n = 1

Population mean

x

Sampling error

(Pop mean - sample mean)

Population

Distribution

Sampling error

n = 2

Population mean

x

Population

Distribution

x

Sampling error


Sample mean

Sampling error

n = 10

Population meanPopulation

Distribution

Sampling error


Sample mean

xxx

x

x

xx

xx x

Generally, as the sample size increases, the sampling error decreases

Generally, as the sample size increases, the sampling error decreases

Sampling error

Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance”

Small population variability Large population variability

Sampling error

These two factors combine to impact the distribution of sample means. The distribution of sample means is a distribution of all possible sample

means of a particular sample size that can be drawn from the population

XA XB XC XD

Population

Samples of size = n

Distribution of sample means

Avg. Sampling error

“chance”

More info

Significance

“A statistically significant difference” means: the researcher is concluding that there is a difference above

and beyond chance with the probability of making a type I error at 5% (assuming an

alpha level = 0.05)

Note “statistical significance” is not the same thing as theoretical significance. Only means that there is a statistical difference Doesn’t mean that it is an important difference

Non-Significance

Failing to reject the null hypothesis Generally, not interested in “accepting the null hypothesis”

(remember we can’t prove things only disprove them) Usually check to see if you made a Type II error (failed to

detect a difference that is really there)• Check the statistical power of your test

• Sample size is too small• Effects that you’re looking for are really small

• Check your controls, maybe too much variability

Summary to this point


H0 is correct

H0 is wrong


Reject H0

Fail to Reject H0

Type I error

Type II error


After treatment period test both groups for memory Results:

Group A’s average memory score is 80% Group B’s is 76%


After treatment period test both groups for memory Results:

Group A’s average memory score is 80% Group B’s is 76%

XAXB

76% 80%


Two sampledistributions

H0: there is no difference between Grp A and Grp B

H0: μA = μB

About populations

“Generic” statistical test

Tests the question: Are there differences between groups

due to a treatment?

One population


H0 is correct

H0 is wrong


Reject H0

Fail to Reject H0

Type I error

Type II error

H0 is true (no treatment effect)

XAXB

76% 80%

Two possibilities in the “real world”

Two sampledistributions


XAXB XAXB

H0 is true (no treatment effect) H0 is false (is a treatment effect)

Two populations


H0 is correct

H0 is wrong


Reject H0

Fail to Reject H0

Type I error

Type II error

76% 80% 76% 80%

People who get the treatment change, they form a new population (the “treatment population)

People who get the treatment change, they form a new population (the “treatment population)

Tests the question: Are there differences between groups

due to a treatment?

Two possibilities in the “real world”


XBXA

Why might the samples be different?(What is the source of the variability between groups)?

ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment


The generic test statistic - is a ratio of sources of variability

Observed difference

Difference from chance=

TR + ID + ER

ID + ER=

Computed test statistic

XBXA

ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment

Sampling error

The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population

XA XB XC XD

Population

Samples of size = n


Avg. Sampling error

“chance”


The generic test statistic distribution To reject the H0, you want a computed test statistics that is large

• reflecting a large Treatment Effect (TR) What’s large enough? The alpha level gives us the decision criterion

Distribution of the test statistic

α-level determines where these boundaries go


Test statistic

TR + ID + ER

ID + ER


Distribution of the test statistic Reject H0

Fail to reject H0




Distribution of the test statistic Reject H0

Fail to reject H0



“One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”)


Things that affect the computed test statistic Size of the treatment effect

• The bigger the effect, the bigger the computed test statistic

Difference expected by chance (sample error)• Sample size• Variability in the population

Significance

“A statistically significant difference” means: the researcher is concluding that there is a difference above

and beyond chance with the probability of making a type I error at 5% (assuming an

alpha level = 0.05)

Note “statistical significance” is not the same thing as theoretical significance. Only means that there is a statistical difference Doesn’t mean that it is an important difference

Non-Significance

Failing to reject the null hypothesis Generally, not interested in “accepting the null hypothesis”

(remember we can’t prove things only disprove them) Usually check to see if you made a Type II error (failed to

detect a difference that is really there)• Check the statistical power of your test

• Sample size is too small• Effects that you’re looking for are really small

• Check your controls, maybe too much variability

Some inferential statistical tests

1 factor with two groups T-tests

• Between groups: 2-independent samples

• Within groups: Repeated measures samples (matched, related)

1 factor with more than two groups Analysis of Variance (ANOVA) (either between groups or

repeated measures)

Multi-factorial Factorial ANOVA

T-test

Design 2 separate experimental conditions Degrees of freedom

• Based on the size of the sample and the kind of t-test

Formula:

T = X1 - X2

Diff by chance

Based on sample error

Observed difference

Computation differs for between and within t-tests

T-test

Reporting your results The observed difference between conditions Kind of t-test Computed T-statistic Degrees of freedom for the test The “p-value” of the test

“The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.”

“The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.”

Analysis of Variance

Designs More than two groups

• 1 Factor ANOVA, Factorial ANOVA• Both Within and Between Groups Factors

Test statistic is an F-ratio Degrees of freedom

Several to keep track of The number of them depends on the design

XBXA XC

Analysis of Variance

More than two groups Now we can’t just compute a simple difference score since

there are more than one difference So we use variance instead of simply the difference

• Variance is essentially an average difference

Observed variance

Variance from chanceF-ratio =

XBXA XC

1 factor ANOVA

1 Factor, with more than two levels Now we can’t just compute a simple difference score since

there are more than one difference• A - B, B - C, & A - C

XBXA XC

1 factor ANOVA

Null hypothesis: H0: all the groups are equal

XA = XB = XC

Alternative hypotheses

HA: not all the groups are equal

XA ≠ XB ≠ XC XA ≠ XB = XC

XA = XB ≠ XC XA = XC ≠ XB

The ANOVA tests this one!!

Do further tests to pick between these

XBXA XC

1 factor ANOVA

Planned contrasts and post-hoc tests:

- Further tests used to rule out the different Alternative

hypothesesXA ≠ XB ≠ XC

XA ≠ XB = XC

XA = XB ≠ XC

XA = XC ≠ XB

Test 1: A ≠ B

Test 2: A ≠ C

Test 3: B = C

Reporting your results The observed differences Kind of test Computed F-ratio Degrees of freedom for the test The “p-value” of the test Any post-hoc or planned comparison results

“The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p <0.05). Groups B and C did not differ significantly from one another”

1 factor ANOVA

Factorial ANOVAs

We covered much of this in our experimental design lecture More than one factor

Factors may be within or between Overall design may be entirely within, entirely between, or mixed

Many F-ratios may be computed An F-ratio is computed to test the main effect of each factor An F-ratio is computed to test each of the potential interactions

between the factors

Factorial ANOVAs

Reporting your results The observed differences

• Because there may be a lot of these, may present them in a table instead of directly in the text

Kind of design• e.g. “2 x 2 completely between factorial design”

Computed F-ratios• May see separate paragraphs for each factor, and for interactions

Degrees of freedom for the test• Each F-ratio will have its own set of df’s

The “p-value” of the test• May want to just say “all tests were tested with an alpha level of

0.05” Any post-hoc or planned comparison results

• Typically only the theoretically interesting comparisons are presented


The following slides are available for a little more concrete review of distribution of sample means discussion.


Distribution of sample means is a “theoretical” distribution between the sample and population

Population Distribution of sample means Sample

• Mean of a group of scores– Comparison distribution is

distribution of means


A simple case Population:

– All possible samples of size n = 2

2 4 6 8

Assumption: sampling with replacement


A simple case Population:

– All possible samples of size n = 2

2 4 6 8

2

4

62

2

82

2

4 4

4

6

8

28

8

8

8

84

64

6

6

6

6

4

6

8

2

4 2

mean mean mean2

3

4

5

3

4

5

6

4

5

6

7

5

6

7

8

There are 16 of them


2

4

6

8

2

4

6

8

2

4 6

2

6

2

6

4 6

4

6

8

28

8

8

8

4

4

4

6

8

2

2

mean mean mean2

3

4

5

3

4

5

6

4

5

6

7

5

6

7

8

means2 3 4 5 6 7 8

5

234

1

In long run, the random selection of tiles leads to a predictable pattern

Properties of the distribution of sample means

Shape If population is Normal, then the dist of sample means

will be Normal

Population Distribution of sample means

N > 30

– If the sample size is large (n > 30), regardless of shape of the population


The mean of the dist of sample means is equal to the mean of the population

Population Distribution of sample means

same numeric valuedifferent conceptual values

• Center


Center The mean of the dist of sample means is equal to the

mean of the population Consider our earlier example

2 4 6 8

Population

μ = 2 + 4 + 6 + 84

= 5


means2 3 4 5 6 7 8

5

234

1

2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+816

=

= 5


Spread• Standard deviation of the population

• Sample size

Putting them together we get the standard deviation of the distribution of sample means

– Commonly called the standard error

Standard error

The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean In other words, it is an estimate of the error that you’d

expect by chance (or by sampling)

All three of these properties are combined to form the Central Limit Theorem– For any population with mean μ and standard

deviation σ, the distribution of sample means for sample size n will approach a normal distribution with a mean of and a standard deviation of as n approaches infinity

(good approximation if n > 30).

Central Limit Theorem


Keep your distributions straight by taking care with your notation

Sample

s

X

Population

σ

μ


Back

statistics (cont.) psych 231: research methods in psychology

Documents

group bs

mean of group bor

mean of group balternative

samplestakes sampling

test statistics step

null hypothesisstep

null hypothesisfail

h0testing hypothesesstep