Statistics review

Basic concepts:

• Variability measures

• Distributions

• Hypotheses

• Types of error

Common analyses

• T-tests

• One-way ANOVA

• Randomized block ANOVA

• Two-way ANOVA

Asks: do two samples come from different populations?

The t-test

A B

DATA

NO YES

Ho

Depends on whether the difference between samples is much greater than difference

within sample.

The t-test

A B

A B

Between >> within…

Depends on whether the difference between samples is much greater than difference

within sample.

The t-test

A B

A B

Between < within…

T-statistic=

Difference between meansStandard error within each sample

The t-test

s2 + s2

n1 n2

How many degrees of freedom?

The t-test

s2 + s2

n1 n2

(n1-1) + (n2-1)

Why does this seem familiar?

T-tables

v 0.10 0.05 0.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

4 1.533 2.132 2.776

infinity 1.282 1.645 1.960

Careful! This table built for one-tailed tests. Only common stats table where to do a two-tailed test (A

doesn’t equal B) requires you to divide the alpha by 2

T-tables

v 0.10 0.05 0.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

4 1.533 2.132 2.776

infinity 1.282 1.645 1.960

Two samples, each n=3, with t-statistic of 2.50: significantly different?

T-tables

v 0.10 0.05 0.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

4 1.533 2.132 2.776

infinity 1.282 1.645 1.960

Two samples, each n=3, with t-statistic of 2.50: significantly different? No!

v 0.10 0.05 0.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

4 1.533 2.132 2.776

infinity 1.282 1.645 1.960

If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap?

v 0.10 0.05 0.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

4 1.533 2.132 2.776

infinity 1.282 1.645 1.960

If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap?

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• the difference in means < 2 x S.E., i.e. t-statistic < 2• and, for any df, t must be > 1.96 to be significant!Car

eful!

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ay a

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General form of the t-test, can have more than 2 samples

One-way ANOVA

Ho:All samples the same…

Ha:At least one sample different

General form of the t-test, can have more than 2 samples

One-way ANOVA

A B C

AB C

A BC

A BC

DATAHo Ha

Just like t-test, compares differences between samples to differences within samples

One-way ANOVA

A B C

Difference between meansStandard error within sample

MS between groupsMS within group

T-test statistic (t)

ANOVA statistic (F)

MS= Sum of squares df

Mean squares:

Everyone gets a lot of cake (high MS) when:

Lots of cake (high SS)

Few forks (low df)

MS= Sum of squares df

MS= Sum of squares df

Mean squares:

Analogous tovariance

Variance:

S2= Σ (xi – x )2

n-1

Sum of squared differences

df

ANOVA tablesdf SS MS F p

Treatment

(between groups)

df (X) SSX

Error

(within groups)

df (E) SSE

Total df (T) SST

SST = SSX SSE

ANOVA tablesdf SS MS F p

Treatment

(between groups)

df (X) SSX SSX

df (X)

Error

(within groups)

df (E) SSE SSE

df (E)

Total df (T) SST

SSXSSE

df (X) df (E)

MSX = = MSE

ANOVA tablesdf SS MS F p

Treatment

(between groups)

df (X) SSX SSX

df (X)

MSX

MSE

Look

up !

Error

(within groups)

df (E) SSE SSE

df (E)

Total df (T) SST

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SSXSSE

df (X) df (E)

MSX = = MSE

Do three species of palms differ in growth rate? We have 5 observations per species. Complete the table!

df SS MS F p

Treatment

(between groups)

69

Error

(within groups)

k(n-1)

Total 104

Hint: For the total df, remember that we calculate total SS as if there are no groups (total variance)…

df SS MS F p

Treatment

(between groups)

69

Error

(within groups)

k(n-1)

Total 104

Note: treatment df always k-1

Is it significant? At alpha = 0.05, F2,12 = 3.89

df SS MS F p

Treatment

(between groups)

2 69 34.5 11.8 ?

Error

(within groups)

12 35 2.92

Total 14 104

2. Randomized block

Good patch Medium patch Poor patch

BLOCK A BLOCK B BLOCK C

Error

Treatment

Error

Treatment

Block

Pro: Can remove between-block SS from error SS…may increase power of test

Error

Treatment

Error

Treatment

Block

Con: Blocks use up error degrees of freedom

Error

Treatment

Error

Treatment

Block

Do the benefits outweigh the costs? Does MS error go down?

F = Treatment SS/treatment dfError SS/error df

Just like one-way ANOVA, except subdivides the treatment SS into:

• Treatment 1• Treatment 2

• Interaction 1&2

Two-way ANOVA

Suppose we wanted to know if moss grows thicker on north or south side of trees, and we look at 10 aspen and 10 fir trees:

• Aspect (2 levels, so 1 df)

• Tree species (2 levels, so 1 df)

• Aspect x species interaction (1df x 1df = 1df)

• Error?

Two-way ANOVA

k(n-1) = 4 (10-1) = 36

v df SS MS F

Aspect 1 SS(Aspect) MS(Aspect) MS(As)

MSE

Species 1 SS(Species) MS(Species) MS(Sp)

MSE

Aspect x Species

1 SS(Int) MS(Int) MS(Int)

MSE

Error

(within groups)

36 SSE MSE

Total 39 SST

Combination of treatments gives non-additive effect

Interactions

Additive effect:

North South

Alder

Fir3 2

5

Combination of treatments gives non-additive effect

Interactions

North South North South

Anything not parallel!

If you log-transformed your variables, the absence of interaction is a multiplicative effect:

log (a) + log (b) = log (ab)

Careful!

North SouthNorth South

Log

(y) y