Means Tests
MARE 250
Dr. Jason Turner
Means Tests
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
Parametric vs. Nonparametric
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)
Means Tests
Parametric – has strict assumptions
1-Sample t-test
2-Sample t-test
Pooled t-test
Non-pooled t-test
Paired t-test
Non-parametric – no assumptions
1-Sample Wilcoxon
2 Sample t-test (Mann-Whitney)
When to Parametric
1 or 2-Sample t-test:
1. Requires large sample size (n=3)
2. Requires normally distributed data
3. 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 Nonparametric
Non-parametric 1 Sample t-test
(Wilcoxon) or 2 Sample t-test (MannWhitney):
1. Small sample size ok
2. Does not require normally distributed
data
3. 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 parametric
2) Fail assumptions tests (Normality,
Equal Variance)
3) Fail to transform (rescale) data to meet
assumptions
Tests with One Mean
Parametric
1-Sample t-test
3 assumptions (not equal variance)
Nonparametric
Wilcoxon test
Tests with One Mean
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
Want to test whether Yellowfin tuna have
levels of Hg below 1.0 ppm
Blue marlin
8.3
FDA
(1.0)
Mako shark
Wahoo
King mackerel
Cobia
Little tunny
Warsaw grouper
Gag grouper
Greater amberjack
Key
Blackfin tuna
Mean
SD
Yellowfin tuna
Dolphin
0
1
2
3
Hg (ppm)
4
5
10
15
Tests with One Mean
H0: μAhi Hg = 1.0ppm
H0: μAhi Hg ≠ 1.0ppm
Tests with One Mean
Parametric
1-Sample t-test – uses the mean and
variance (std. dev.) of raw data
Nonparametric
Wilcoxon test – uses the median of the raw
data
Tests with Two Means
Parametric - require 4 assumptions
2-Sample t-test
Pooled
Unpooled
Paired
Nonparametric
Mann-Whitney test
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*
Tests with Two Means
Compare 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
Which Test to Run
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*
Paired T-test
Parametric t-tests – data not independent
Paired t-test
For example:
Growth study on mark-recaptured ahi
July 2011
July 2012
Paired T-test
Parametric t-tests
Paired 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
Which Test to Run
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*
Normality Test
H0 hypothesis: data
normally distributed
Probability Plot of Weight
Normal
99.9
Mean
StDev
N
RJ
P-Value
99
95
Percent
90
80
70
60
50
40
30
20
10
5
1
0.1
-200
-100
0
100
200
Weight
300
192.2
110.5
143
0.955
<0.010
If p value is less than α,
then reject H0
Data does not follow a
400
500 distribution
600
normal
Mann-Whitney T-test
Nonparametric t-tests
Mann-Whitney
Compare medians from two groups of raw
data
H0: μurchins deep = μurchins shallow
Ha: μurchins deep ≠ μurchins shallow
Which Test to Run
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-tests
Non-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 ttest
Which Test to Run
Parametric t-tests
Pooled 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
When Do I Do The What Now?
“Well, whenever I'm confused, I just check my underwear. It holds the answer
to all the important questions.” – Grandpa Simpson
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
When Do I Do The What Now?
“Well, whenever I'm confused, I just check my underwear. It holds the answer
to all the important questions.” – Grandpa Simpson
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 (MannWhitney)