Statistical Tests

Data Analysis

Statistics - a powerful tool for analyzing data

1. Descriptive Statistics - provide an overview of the attributes of a data set. These include measurements of central tendency (frequency histograms, mean, median, & mode) and dispersion (range, variance & standard deviation)

2. Inferential Statistics - provide measures of how well your data support your hypothesis and if your data are generalizable beyond what was tested (significance tests)

Inferential Statistics

2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 107 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 79 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 59 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 105 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 44 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 57 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 103 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 21 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 15 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 76 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 31 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 44 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 25 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 38 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 95 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 61 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 83 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 101 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 38 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 64 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2

The Population: =5.314

Population size = 500

2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 10

7 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 7

9 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 5

9 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 105 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 44 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 5

7 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 10

3 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 21 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 1

5 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 7

6 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 31 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 44 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 25 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 38 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 9

5 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 6

1 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 8

3 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 10

1 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 3

8 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 64 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2

The Sample: 7, 6, 4, 9, 8, 3, 2, 6, 1mean = 5.111


2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 107 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 7

9 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 5

9 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 10

5 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 4

4 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 57 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 10

3 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 2

1 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 15 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 7

6 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 3

1 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 4

4 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 2

5 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 3

8 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 95 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 61 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 83 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 10

1 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 3

8 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 6

4 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2

The Sample: 1, 5, 8, 7, 4, 1, 6, 6mean = 4.75


Parametric or Non-parametric?

•Parametric tests are restricted to data that: 1) show a normal distribution 2) * are independent of one another 3) * are on the same continuous scale of measurement

•Non-parametric tests are used on data that: 1) show an other-than normal distribution 2) are dependent or conditional on one another 3) in general, do not have a continuous scale of measurement

e.g., the length and weight of something –> parametric vs. did the bacteria grow or not grow –> non-parametric

The First Question

After examining your data, ask: does what you're testingseem to be a question of relatedness or a question ofdifference?

If relatedness (between your control and your experimentalsamples or between you dependent and independent variable), you will be using tests for correlation (positive or negative) or regression.

If difference (your control differs from your experimental),you will be testing for independence between distributions,means or variances. Different tests will be employed ifyour data show parametric or non-parametric properties.

See Flow Chart on page 50 of HBI.

Tests for Differences

• Between Means - t-Test - P - ANOVA - P - Friedman Test - Kruskal-Wallis Test - Sign Test - Rank Sum Test• Between Distributions - Chi-square for goodness of fit - Chi-square for independence• Between Variances - F-Test – PP – parametric tests

Differences Between Means

Asks whether samples come from populations with different means

Null Hypothesis Alternative Hypothesis

A

Y

B CA

Y

B C

There are different tests if you have 2 vs more than 2 samples

Differences Between Means – Parametric Data

t-Tests compare the means of two parametric samples

E.g. Is there a difference in the mean height of men and women?

HBI: t-Test

Excel: t-Test (paired and unpaired) – in Tools – Data Analysis

A researcher compared the height of plants grown in high and low light levels. Her results are shown below. Use a T-test to determine whether there is a statistically significant difference in the heights of the two groups

Low Light High Light49 4531 4043 5931 5840 5544 5049 4648 5333 43

Differences Between Means – Parametric Data

ANOVA (Analysis of Variance) compares the means of two or more parametric samples.

E.g. Is there a difference in the mean height of plants grown under red, green and blue light?

HBI: ANOVA

Excel: ANOVA – check type under Tools – Data Analysis

weight of pigs fed different foods

food 1 food 2 food 3 food 4

60.8 68.7 102.6 87.9

57.0 67.7 102.1 84.2

65.0 74.0 100.2 83.1

58.6 66.3 96.5 85.7

61.7 69.8 90.3

A researcher fed pigs on four different foods. At the endof a month feeding, he weighed the pigs. Use an ANOVAtest to determine if the different foods resulted indifferences in growth of the pigs.

Correlations look for relationships between two variables which may not be functionally related. The variables may be ordinal, interval, or ratio scale data. Remember, correlation does not prove causation; thus there may not be a cause and effect relationship between the variables.

E.g. Do species of birds with longer wings also have longer necks?

HBI: Spearman’s Rank Correlation (NP)

Excel: Correlation (P)

Correlation

Question – is there a relationship between students aptitude for mathemathics and for biology?

Student Math score Math Rank Biol. score Biology rank

1 57 3 83 7

2 45 1 37 1

3 72 7 41 2

4 78 8 84 8

5 53 2 56 3

6 63 5 85 9

7 86 9 77 6

8 98 10 87 10

9 59 4 70 5

10 71 6 59 4

Regressions look for functional relationships between two continuous variables. A regression assumes that a change in X causes a change in Y.

E.g. Does an increase in light intensity cause an increase in plant growth?

HBI: Regression Analysis (P)

Excel: Regression (P)

Regression

Correlation & Regression

Looks for relationships between two continuous variables


X

Y

X

Y

Is there a relationship between wing length and tail length in songbirds?

wing length cm tail length cm

10.4 7.4

10.8 7.6

11.1 7.9

10.2 7.2

10.3 7.4

10.2 7.1

10.7 7.4

10.5 7.2

10.8 7.8

11.2 7.7

10.6 7.8

11.4 8.3

Is there a relationship between age and systolic blood pressure?

Age (yr)

systolic blood pressuremm hg

30 108

30 110

30 106

40 125

40 120

40 118

40 119

50 132

50 137

50 134

60 148

60 151

60 146

60 147

60 144

70 162

70 156

70 164

70 158

70 159

Chi square tests compare observed frequency distributions, either to theoretical expectations or to other observed frequency distributions.

Differences Between Distributions


E.g. The F2 generation of a cross between a round pea and a wrinkled pea produced 72 round individuals and 20 wrinkled individuals. Does this differ from the expected 3:1 round:wrinkled ratio of a simple dominant trait?

HBI: Chi-Square One Sample Test (goodness of fit)Excel: Chitest – under Function Key – Statistical

Smooth

Fre

quen

cy

Wrinkled

E

E

E.g. 67 out of 100 seeds placed in plain water germinated while 36 out of 100 seeds placed in “acid rain” water germinated. Is there a difference in the germination rate?

HBI: Chi-Square Two or More Sample Test (independence)Excel: Chitest – under Function key - Statistical

Plain Acid PlainP

ropo

rtio

nG

erm

inat

ion

Acid

Pro

port

ion

Ger

min

atio

n

Null HypothesisAlternative Hypothesis



Are there differences in the numbers of male and female Cardinals at different times of the year?

Sex Spring Summer Fall Winter

Male 163 135 71 43

Female 86 77 40 38

This is a special case for a Chi-Square Test known as a Contingency Table (as was the previous example)

Differences Between Means

Asks whether samples come from populations with different means


A

Y

B CA

Y

B C

There are different tests if you have 2 vs more than 2 samples

Aplysia punctata – the sea hare

Aplysia parts

Differences Between Means – Non-Parametric Data

The Sign Test compares the means of two “paired”, non-parametric samples

E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once at night and once during the day –> paired data.

HBI: Sign Test

Excel: N/A

SubjectNight

ResponseDay

Response1 2 52 1 33 2 2

The Friedman Test is like the Sign test, (compares the means of “paired”, non-parametric samples) for more than two samples.

E.g. Is there a difference in the gill withdrawal response of Aplysia between morning, afternoon and evening? Each subject has been tested once during each time period –> paired data

HBI: Friedman Test

Excel: N/A

SubjectMorning

ResponseAfternoon Response

Evening. Response

1 4 3 22 5 2 13 3 4 3


The Rank Sum test compares the means of two non-parametric samples

E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once, either during the night or during the day –> unpaired data.

HBI: Rank Sum

Excel: N/A

SubjectNight

ResponseDay

Response1 52 13 24 35 46 17 5


The Kruskal-Wallis Test compares the means of more than two non-parametric, non-paired samples

E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once, either during the morning, afternoon or evening –> unpaired data.

HBI: Kruskal-Wallis Test

Excel: N/A


SubjectMorning

ResponseAfternoon Response

Evening. Response

1 42 53 44 35 26 3

Statistical Tests

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Transcript of Statistical Tests