One-Way Analysis of Variance (ANOVA)

Aside

I dislike the ANOVA chapter (11) in the book with a passion – if it were all written like this I wouldn’t use the thing

Therefore, don’t worry if you don’t understand it, use my notes – Exam questions on ANOVA will come from my lectures and notes

One-Way ANOVA

One-Way Analysis of Variance aka One-Way ANOVA Most widely used statistical technique in all of

statistics One-Way refers to the fact that only one IV and

one DV are being analyzed (like the t-test) i.e. An independent-samples t-test with treatment

and control groups where the treatment (present in the tx grp and absent in the control grp) is the IV

One-Way ANOVA

Unlike the t-test, the ANOVA can look at levels or subgroups of IV’s The t-test can only test if an IV is there or not, not

differences between subgroups of the IV I.e. our experiment is to test the effect of hair color (our

IV) on intelligence One t-test can only test if brunettes are smarter than

blondes, any other comparison would involve doing another t-test

A one-way ANOVA can test many subgroups or levels of our IV “hair color”, for instance blondes, brunettes, and redheads are all subtypes of hair color, can so can be tested with one one-way ANOVA

One-Way ANOVA

Other examples of subgroups: If “race” is your IV, then caucasian, african-american,

asian-american, hispanic (4) are all subgroups/levels If “gender” is your IV, than male and female (2) are

your levels If “treatment” is your IV, then some treatment, a little

treatment, and a lot of treatment (3) can be your levels

One-Way ANOVA

OK, so why not just do a lot of t-tests and keep things simple?

1. Many t-tests will inflate our Type I Error rate! This is an example of using many statistical tests to

evaluate one hypothesis – see: the Bonferroni Correction

2. It is less time consuming There is a simple way to do the same thing in ANOVA,

they are called post-hoc tests, and we will go over them later on

However, with only one DV and one IV (with only two levels), the ANOVA and t-test are mathematically identical, since they are essentially derived from the same source

One-Way ANOVA

Therefore, the ANOVA and the t-test have similar assumptions: Assumption of Normality

Like the t-test you can place fast and loose with this one, especially with large enough sample size – see: the Central Limit Theorem

Assumption of Homogeneity of Variance (Homoscedasticity)

Like the t-test, this isn’t problematic unless one level’s variance is much larger than one the others (~4 times as large) – the one-way ANOVA is robust to small violations of this assumption, so long as group size is roughly equal

One-Way ANOVA

Independence of Observations Like the t-test, the ANOVA is very sensitive to

violations of this assumption – if violated it is more appropriate to use a Repeated-Measures ANOVA

One-Way ANOVA

Hypothesis testing in ANOVA: Since ANOVA tests for differences between

means for multiple groups or levels of our IV, then H1 is that there is a difference somewhere between these group means

H1 = μ1 ≠ μ2 ≠ μ3 ≠ μ4, etc…

Ho = μ1 = μ2 = μ3 = μ4, etc… These hypothesis are called omnibus hypothesis,

and tests of these hypotheses omnibus tests

One-Way ANOVA

However, our F-statistic does not tell us where this difference lies If we have 4 groups, group 1 could differ from groups 2-4,

groups 2 and 4 could differ from groups 1 and 3, group 1 and 2 could differ from 3, but not 4, etc.

Since our hypothesis should be as precise as possible (presuming you’re researching something that isn’t completely new), you will want to determine the precise nature of these differences You can do this using multiple comparison techniques (more

on this later)

One-Way ANOVA

The basic logic behind the ANOVA:

The ANOVA yields and F-statistic (just like the t-test gave us a t-statistic)

The basic form of the F-statistic is: MStreatment/MSerror

MS = mean square or the mean of squares (why it is called this will be more obvious later)

One-Way ANOVA

The basic logic behind the ANOVA: MSbetween or MStreatment = average variability

(variance) between the levels of our IV/groups Ideally we want to maximize MStreatment, because

we’re predicting that our IV will differentially effect our groups

i.e. if our IV is treatment, and the levels are no treatment vs. a lot of treatment, we would expect the treatment group mean to be very different than the no treatment mean – this results in lots of variability between these groups

One-Way ANOVA

The basic logic behind the ANOVA: MSwithin or MSerror = average variance among subjects

in the same group• Ideally we want to minimize MSerror, because ideally our

IV (treatment) influences everyone equally – everyone improves, and does so at the same rate (i.e. variability is low)

If F = MStreatment/ MSerror, then making MStreatment

large and MSerror small will result in a large value of F

Like t, a large value corresponds to small p-values, which makes it more likely to reject Ho

One-Way ANOVA

However, before we calculate MS, we need to calculate what are called sums of squares, or SS SS = the sum of squared deviations around the

mean Does this sound familiar? What does this sound

like? Just like MS, we have SSerror and SStreatment

Unlike MS, we also have SStotal = SSerror + SStreatment

One-Way ANOVA

SStotal = Σ(Xij - )2 =

It’s the formula for our old friend variance, minus the n-1 denominator!

Note: N = the number of subjects in all of the groups added together

..X N

XX

22

One-Way ANOVA

SStreatment =

This means we:1. Subtract the grand mean, or the mean of all of the

individual data points, from each group mean

2. Square these numbers

3. Multiply them by the number of subjects from that particular group

4. Sum themNote: n = number of subjects per groupHint: The number of numbers that you sum should equal the

number of groups

2..XXn jj

One-Way ANOVA

That leaves us with SSerror = SStotal – SStreatment

Remember: SStotal = SSerror + SStreatment

Degrees of freedom: Just as we have SStotal,SSerror, and SStreatment, we also have

dftotal, dferror, and dftreatment

dftotal = N – 1 OR the total number of subjects in all groups minus 1

dftreatment = k – 1 OR the number of levels of our IV (aka groups) minus 1

dferror = N – k OR the total number of subjects minus the number of groups OR dftotal - dftreatment

One-Way ANOVA

Now that we have our SS and df, we can calculate MS MStreatment = SStreatment/dftreatment

MSerror = SSerror/dferror

Remember: MSbetween or MStreatment = average variability

(variance) between the levels of our IV/groups MSwithin or MSerror = average variance among

subjects in the same group

One-Way ANOVA

We then use this to calculate our F-statistic: F = MStreatment/ MSerror

The p-value associated with this F-statistic is a function of both F and your df Higher F and/or df Lower p Recall: df = n & # levels of your IV

More S’s and/or fewer levels Higher df Lower p

One-Way ANOVA

How can we change our experiment to increase the likelihood of a significant result/decrease p? Larger ES Higher F:

Increase the potency of the IV Higher df

More S’s Fewer # levels of your IV

• Collapse across groups – Instead of looking at Kids vs. Young Adults vs. Adults, look at Children vs. Adults only

• Worst way to decrease p as this sacrifices how subtle-ly you can test your theory

One-Way ANOVA

Example: What effect does smoking have on

performance? Spilich, June, and Renner (1992) asked nonsmokers (NS), smokers who had delayed smoking for three hours (DS), and smokers who were actively smoking (AS) to perform a pattern recognition task in which they had to locate a target on a screen. The data follow:

One-Way ANOVA

Example:

1. What is the IV, number of levels, and the DV?

2. What is H1 and Ho?

3. What is your dftotal, dfgroups, and dferror?

Non-Smokers Delayed Smokers

Active Smokers

9 12 8

8 7 8

12 14 9

10 4 1

7 8 9

10 11 7

9 16 16

11 17 19

8 5 1

10 6 1

8 9 22

10 6 12

8 6 18

11 7 8

10 16 10

One-Way ANOVA

Descriptives

PERFORMA

15 9.4000 1.40408 .36253 8.6224 10.1776 7.00 12.00

15 9.6000 4.40454 1.13725 7.1608 12.0392 4.00 17.00

15 9.9333 6.51884 1.68316 6.3233 13.5433 1.00 22.00

45 9.6444 4.51339 .67282 8.2885 11.0004 1.00 22.00

ns

ds

as

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

ANOVA

PERFORMA

2.178 2 1.089 .051 .950

894.133 42 21.289

896.311 44

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

One-Way ANOVA

Example:

4. Based on these results, would you reject or fail to reject Ho?

5. What conclusion(s) would you reach about the effect of the IV on the DV?

One-Way ANOVA

Assumptions of ANOVA: Independence of Observations Homoscedasticity Normality

Equal sample sizes – not technically an assumption, but effects the other 3

How do we know if we violate one (or more) of these? What do we do?

One-Way ANOVA

Independence of Observations Identified methodologically Other than using repeated-measures tests

(covered later), nothing you can do

Equal Sample Sizes Add more S’s to the smaller group DON’T delete S’s from the larger one

One-Way ANOVA

Homoscedasticity Identified using Levene’s Test or the Welch

Procedure Again, don’t sweat the book, SPSS will do it for

you If detected (and group sizes very unequal), use

appropriate transformation

One-Way ANOVA

Homoscedasticity

Descriptives

6 2.714

6 1.329

12 2.067

6 2.098

6 2.828

12 2.431

6 2.714

6 2.338

12 2.417

6 1.835

6 3.445

12 2.864

1

2

Total

1

2

Total

1

2

Total

1

2

Total

Trial 1

Trial 2

Trial 3

Trial 4

N Std. Deviation

Test of Homogeneity of Variances

3.312 1 10 .099

.156 1 10 .701

.266 1 10 .617

7.788 1 10 .019

Trial 1

Trial 2

Trial 3

Trial 4

LeveneStatistic df1 df2 Sig.

Robust Tests of Equality of Means

.292 1 7.268 .605

.484 1 9.223 .504

.013 1 9.786 .912

1.849 1 7.626 .213

Welch

Welch

Welch

Welch

Trial 1

Trial 2

Trial 3

Trial 4

Statistica

df1 df2 Sig.

Asymptotically F distributed.a.

One-Way ANOVA

Normality Can identify with histograms of DV’s (IV’s are

supposed to be non-normal) More appropriate to use skewness and kurtosis

statistics If detected (and sample size very small), use

appropriate transformation

One-Way ANOVA

Normality1. Divide statistic by its standard error to get z-score

2. Calculate p-value using z-score and df = n

Descriptive Statistics

12 12 19 16.50 2.067 -.815 .637 .651 1.232

12 8 16 11.50 2.431 .205 .637 -.406 1.232

12 4 12 7.75 2.417 .165 .637 -.864 1.232

12 1 9 4.25 2.864 .534 .637 -1.192 1.232

12

Trial 1

Trial 2

Trial 3

Trial 4

Valid N (listwise)

Statistic Statistic Statistic Statistic Statistic Statistic Std. Error Statistic Std. Error

N Minimum Maximum Mean Std.Deviation

Skewness Kurtosis

One-Way ANOVA

Estimates of Effect Size in ANOVA:1. η2 (eta squared) = SSgroup/SStotal

Unfortunately, this is what most statistical computer packages give you, because it is simple to calculate, but seriously overestimates the size of effect

2. ω2 (omega squared) =

Less biased than η2, but still not ideal

errortotal

errorgroups

MSSS

MSkSS

1

One-Way ANOVA

Estimates of Effect Size in ANOVA:3. Cohen’s d =

Remember: for d, .2 = small effect, .5 = medium, and .8 = large

2

22

21

21

nn

t

df

F

s

XX

errorp

One-Way ANOVA

Multiple Comparison Techniques: Remember: ANOVA tests for differences

somewhere between group means, but doesn’t say where

H1 = μ1 ≠ μ2 ≠ μ3 ≠ μ4, etc… If significant, group 1 could be different from groups

2-4, groups 2 & 3 could be different from groups 1 and 4, etc.

Multiple comparison techniques attempt to detect specifically where these differences lie

Multiple Comparison Techniques

You could always run 2-sample t-tests on all possible 2-group combinations of your groups, although with our 4 group example this is 6 different tests

Running 6 tests @ (α = .05) = (α = .05 x 6 = .3) This would inflate what is called the familywise error rate

– in our previous example, all of the 6 tests that we run are considered a family of tests, and the familywise error rate is the α for all 6 tests combined (.3) – however, we want to keep this at .05

Multiple Comparison Techniques

To perform multiple comparisons with a significant omnibus F, or not to …

Why would you look for a difference between two or more groups when your F said there isn’t one?

Multiple Comparison Techniques

Some say this is what is called statistical fishing and is very bad – you should not be conducting statistical tests willy-nilly without just cause or a theoretical reason for doing so

Think of someone fishing in a lake, you don’t know if anything is there, but you’ll keep trying until you find something – the idea is that if your hypothesis is true, you shouldn’t have to look to hard to find it, because if you look for anything hard enough you tend to find it

Multiple Comparison Techniques

However, others would say that the omnibus test is underpowered, particularly with a large number of groups and if only a few significant differences are predicted among them

i.e. H1 = μ1 ≠ μ2 ≠ μ3 ≠ μ4 … μ10 If you only predict groups 1 and 5 will differ, you are

unlikely to get a significant omnibus F unless you have a ton of S’s

I and most statisticians fall on this side of the argument – i.e. it’s OK to ignore the omnibus test if you have a theoretical reason to predict specific differences among groups

A Priori Techniques

A priori techniques: Planned prior to data collection Involve specific hypotheses between group means (i.e.

not just testing all group differences) Multiple t Tests

As stated previously, not a good idea bc/ inflated Involves a different formula than a traditional t test

Bonferroni t (Dunn’s Test) Simply using the Bonferroni Correction on a bunch of

regular t tests For 6 comparisons: = .05/6 = .0083 Can combine the Bonferroni correction with the adjusted

formula for t used above

A Priori Techniques

Dunn-Šidák Test Almost identical to the Bonferroni t, but uses 1 – (1 -

)1/c instead of /c, where c = # of comparisons Slightly less conservative For 6 comparisons = 1-(1-.05)1/6 = .0085

A Priori Techniques

Holm Test Instead of assuming we evaluate hypotheses all at once,

allows for ordering of when hypotheses are evaluated Uses most stringent for hypothesis w/ strongest

support (largest t) Adjusts downward for next comparisons, taking into

account that previous comparison was significant More powerful than Bonferroni t & Dunn-Šidák Test

1. Calculate t tests for all comparisons2. Arrange t’s in decreasing order3. For 1st t test, use Dunn-Šidák method with normal c4. For 2nd t test, use c – 15. For 3rd t test, use c – 2, etc.6. Continue until a nonsignificant result is obtained

A Priori Techniques

Linear Contrasts What if, instead of comparing group x to group y, we

want to compare group x, y, & z to group a & b? Coefficients – how to tell mathematically which

groups we are comparing• Coefficients for the same groups have to be the same

and all coefficients must add up to 0• Comparing groups 1, 2, & 3 to groups 4 & 5:

Groups 1 – 3: Coefficient = 2 Groups 4 & 5: Coefficient = -3

Use Bonferroni correction to adjust to # of contrasts• 4 contrasts use = .05/4 = .0125

A Priori Techniques

Orthogonal Contrasts What if you want to compare groups within a

contrast? I.e. Group 1 vs. Groups 2 & 3 and Group 2 vs. Group

3 Assigning coefficients is the same, but calculations

are different (don’t worry about how different, just focus on the linear vs. orthogonal difference)

A Priori Techniques

Both the Holm Test and linear/orthogonal contrasts sound good, which do I use? If making only a few contrasts:

Linear/Orthogonal If making many contrasts: Holm Test

More powerful and determining coefficients is confusing with multiple contrasts

Post Hoc Techniques

Post hoc techniques: Fisher’s LSD

We replace in our 2-sample t-test formula with MSerror, and we get:

We then test this using a critical t, using our t-table and dferror as our df

You can use either a one-tailed or two-tailed test, depending on whether or not you think one mean is higher or lower (one-tailed) or possibly either (two-tailed) than the other

2ps

21

21

11

nnMS

XXt

error

Post Hoc Techniques

Fisher’s LSD However, with more than 3 groups, using Fisher’s

LSD results in an inflation of (i.e. with 4 groups α = .1)

You could use the Bonferroni method to correct for this, but then why not just use it in the first place?

This is why Fisher’s LSD is no longer widely used and other methods are preferred

Newman-Keuls Test Like Fisher’s LSD, allows the familywise > .05 Pretty crappy test for that reason

Post Hoc Techniques

Scheffé’s Test Fishers LSD & Neuman-Keuls = not conservative

enough = too easy to find significant results Scheffé’s Test = too conservative = result in a low

degree of Type I Error but too high Type II Error (incorrectly rejects H1) = too hard to find significant results

Tukey’s Honestly Significant Difference (HSD) test Very popular, but conservative

Post Hoc Techniques

Ryan/Einot/Gabriel/Welsch (REGWQ) Procedure

Like Tukey’s HSD, but adjusts (like the Dunn-Šidák Test) to make the test less conservative

Good compromise of Type I/Type II Error

Dunnett’s Test Specifically designed for comparing a control group

with several treatment groups Most powerful test in this case

One-Way ANOVA

Reporting and Interpreting Results in ANOVA: We report our ANOVA as:

F(dfgroups, dftotal) = x.xx, p = .xx, d = .xx i.e. for F(4, 299) = 1.5, p = .01, d = .01 – We have 5

groups, 300 subjects total in all of our groups put together; We can reject Ho, however our small effect size statistic informs us that it may be our large sample size that resulted in us doing so rather than a large effect of our IV