Means Tests

MARE 250Dr. Jason Turner

Type of stats test called a means test

Tests for differences in samples based upon their average (mean) and standard deviation (variance)

Several versions from 1 sample, 2 sample, through multiple samples

Means Tests

Response – variable of interest; variable you collect - #Fish, %Coral cover, temperature, salinity, etc

Factor – variable by which response is divided; categorical - location, Date, Gender, Species

Level – components of factor; - Location (Puako, Hilo Bay), Date (Jan, Feb), Gender (♂, ♀)

Means Tests

2 Types:

1) Parametric Means tests – have defined assumptions including normally distributed data

2) Nonparametric Means tests – have few/no assumptions

Means Tests

Parametric means tests – require data to be normal, etc (assumptions)

Nonparametric tests – do not require data to be normal (assumptions)

Parametric vs. Nonparametric

Parametric means tests – include 2 sample t-test, ANOVA

Nonparametric means tests – include Mann Whitney (t-test), Kruskal Wallace (ANOVA)

Parametric vs. Nonparametric

Parametric – has strict assumptions1-Sample t-test2-Sample t-test

Pooled t-testNon-pooled t-testPaired t-test

Non-parametric – no assumptions1-Sample Wilcoxon2 Sample t-test (Mann-Whitney)

Means Tests

1 or 2-Sample t-test:

1. Requires large sample size (n=3)2. Requires normally distributed data3. Outliers can significantly confound

results

Parametric testing is the “gold standard” – it is the type of test we attempt first

Has very strict criteria called assumptions

When to Parametric

Has 4 Assumptions:

1. Random Samples – collected randomly

2. Independent Samples – equal chance

3. Normal Populations (or large

samples; n=3)

4. Variances (std. dev.) are equal

When to Parametric

How do we assess these 4 Assumptions:

1. Random Samples – Sampling design

2. Independent Samples – Sampling Design

3. Normal Populations - Normality

test*

4. Variances - Equal Variance test*

When to Parametric

Non-parametric 1 Sample t-test (Wilcoxon) or 2 Sample t-test (Mann-Whitney):

1. Small sample size ok2. Does not require normally distributed

data3. Outliers do not confound results

When to Nonparametric

Non-parametric test are used heavily in some disciplines – although not typically in the natural sciences

Used when data are not normal, or low sample size, low “power”

When to Nonparametric

When do we run Nonparametric tests?

1) Sample size is too small for parametric2) Fail assumptions tests (Normality,

Equal Variance)3) Fail to transform (rescale) data to meet

assumptions

When to Nonparametric

Tests with One MeanParametric 1-Sample t-test

3 assumptions (not equal variance)

NonparametricWilcoxon test

Also called 1 mean t-tests

Compare collected dataset with a value

For example:

The FDA has issued fish consumption advisories for populations containing Hg levels greater than 1.0 ppm.

Tests with One Mean

0 1 2 3 4

Blue marlin

Mako shark

Little tunny

Warsaw grouper

Greater amberjack

Blackfin tuna

Yellowfin tuna

Dolphin

5 10 15

8.3

Key

Mean

FDA(1.0)

Wahoo

King mackerel

Cobia

Gag grouper

Hg (ppm)

SD

Want to test whether Yellowfin tuna have levels of Hg below 1.0 ppm

Tests with One Mean

H0: μAhi Hg = 1.0ppm

H0: μAhi Hg ≠ 1.0ppm

Tests with One Mean

Tests with One MeanParametric 1-Sample t-test – uses the mean and variance (std. dev.) of raw data

NonparametricWilcoxon test – uses the median of the raw data

Tests with Two MeansParametric - require 4 assumptions 2-Sample t-test

PooledUnpooledPaired

NonparametricMann-Whitney test

How do we assess these 4 Assumptions:

1. Random Samples – Sampling design

2. Independent Samples – Sampling Design

3. Normal Populations - Normality

test*

4. Variances - Equal Variance test*

When to Parametric

Tests with Two MeansCompare means from two groups of raw data H0: μurchins deep = μurchins shallow

Ha: μurchins deep ≠ μurchins shallow

Most widely applied statistical tests

Variety of Parametric tests (3)

Single Nonparametric test

How do we assess these 4 Assumptions:

1. Random Samples – Sampling design

2. Independent Samples – Sampling Design

3. Normal Populations - Normality

test*

4. Variances - Equal Variance test*

Which Test to Run

Paired T-testParametric t-tests – data not independentPaired t-test

For example:

Growth study on mark-recaptured ahi

July 2011 July 2012

Paired T-testParametric t-testsPaired t-test

Conduct a paired t-test - If the samples are not independent

Used when there is a natural pairing of the members of two populations

Calculates difference between the two paired samples

How do we assess these 4 Assumptions:

1. Random Samples – Sampling design

2. Independent Samples – Sampling Design

3. Normal Populations - Normality

test*

4. Variances - Equal Variance test*

Which Test to Run

Normality Test

Weight

Perc

ent

6005004003002001000-100-200

99.9

99

95

90

80706050403020

10

5

1

0.1

Mean

<0.010

192.2StDev 110.5N 143RJ 0.955P-Value

Probability Plot of WeightNormal

H0 hypothesis: data normally distributed

If p value is less than α, then reject H0

Data does not follow a normal distribution

Mann-Whitney T-testNonparametric t-testsMann-WhitneyCompare medians from two groups of raw data

H0: μurchins deep = μurchins shallow

Ha: μurchins deep ≠ μurchins shallow

How do we assess these 4 Assumptions:

1. Random Samples – Sampling design

2. Independent Samples – Sampling Design

3. Normal Populations - Normality

test*

4. Variances - Equal Variance test*

Which Test to Run

Parametric t-testsNon-pooled t-test

Conduct a Non-pooled t-test - you cannot “pool” the samples because the variances are not equal

In Minitab – do not check box – “Assume Equal Variances” when running 2-sample t-test

Which Test to Run

Parametric t-testsPooled t-test

Conduct a pooled t-test - you can “pool” the samples because the variances are assumed to be equal

In Minitab - check box – “Assume Equal Variances” when running 2-sample t-test

Which Test to Run

When Do I Do The What Now?

If all 4 assumptions are met:Conduct a pooled t-test - you can “pool” the samples because the variances are assumed to be equal

If the samples are not independent:Conduct a paired t-test

“Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson

When Do I Do The What Now?

If the variances (std. dev.) are not equal:Conduct a non-pooled t-test

If the data is not normal or has small sample size:Conduct a non-parametric t-test (Mann-Whitney)

“Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson