Means Tests MARE 250 Dr. 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 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) Means Tests

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 Parametric

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

Tests with One Mean Parametric 1-Sample t-test 3 assumptions (not equal variance) Nonparametric Wilcoxon 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

Want to test whether Yellowfin tuna have levels of Hg below 1.0 ppm Tests with One Mean

H 0 : μ Ahi Hg = 1.0ppm H 0 : μ Ahi 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

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 Means Compare means from two groups of raw data H 0 : μ urchins deep = μ urchins shallow H a : μ urchins deep ≠ μ urchins 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-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

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 H 0 hypothesis: data normally distributed If p value is less than α, then reject H 0 Data does not follow a normal distribution

Mann-Whitney T-test Nonparametric t-tests Mann-Whitney Compare medians from two groups of raw data H 0 : μ urchins deep = μ urchins shallow H a : μ urchins deep ≠ μ urchins 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-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 t- test 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 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