Hypothesis Testing - Relationships

Updated 6 November 2009

“I’ve learned that you can’t be taken seriously in any scientific discipline without an understanding of statistics.”

Bethany Williams, PhD, U.S. Geological Survey (HU 2004)

You are expected to read the corresponding textbook chapters as we cover them in class.

Working the exercises at the end of the chapters will enhance your understanding.

(click for answers to all exercises)

Analysis of relationships (among cases of variables)

A. Correlation (=association); no functional dependence (e.g., wing length vs. tail length)

B. Example: positive, negative, and no relationship

C. 2 questions:

1. are two variables related in a linear way? (significant? =reject null of no relationship)

a. if answer is “no”, do not go further

b. if answer is “yes”, ask:

2. what is the strength of the relationship? (value of correlation coefficient)

Statistical Tests
		Parametric	Nonparametric	Assumptions of parametric tests · Data randomly sampled and independent · Data measured on ratio/interval scale · Variables continuous or discrete (if n and no. possible values large) · Data normally distributed for each group (residuals in ANOVA/regression) · For questions regarding means, data variances among groups (residuals in ANOVA/regression) homogeneous Assumptions of non-parametric tests · Data randomly sampled and independent	Guidelines for determining appropriate analyses · Read the question carefully; make sure you understand what question is asking · Look for key words: difference, relationship, association, correlation · A “v” word, (vary, variance, variation) will be present for questions of differences in variances · If a “v” word does not appear in a difference question and question does not concern frequencies, assume question concerns means · The word “affect” (or “effect”) can be used in both difference and relationship questions - must understand use in context
Differences	Means (2)	Indep. samples t	Mann-Whitney
		Paired samples t	Wilcoxon
	Means	ANOVA	Kruskal-Wallis
	Variances	Bartlett’s	Levene’s
	Frequencies	-----	Goodness-of-fit
Relationships	Cases	Pearson correlation	Spearman correlation
		Regression	-----
	Frequencies	-----	Independence

Pearson correlation (Chap. 13)

1. Purpose: Test whether the cases of two variables are correlated

2. Comments: if variables are related, the relationship is linear

3. Null hypothesis: H₀: r_{(var1, var2)} = 0

4. Test statistic (correlation coefficient, r (varies from -1 to +1) and probability source: Systat/Systat

5. r² (coefficient of determination) - proportion of variation in Y that is explained by variation in X

6. SYSTAT path: Analyze®Correlation®Simple (enter variables; Continuous Data)

SYSTAT output: (EGGSIZE.SYD; select sex=1 and year=81); pic

Pearson correlation matrix

HWGT HSVL

HWGT 1.0000

HSVL 0.8297 1.0000

Bartlett Chi-square statistic: 36.735 DF=1 Prob= 0.000

Matrix of Probabilities

HWGT HSVL

HWGT 0.0000

HSVL 0.0000 0.0000

Number of observations: 34

Bonferroni probabilities - use when correlating >2 variables from one data set
(multiple comparisons increase chances of Type I error – must adjust probabilities)

SYSTAT output: (SHRIMPS.SYD); pic
 
Kolmogorov-Smirnov One Sample Test using Normal(0.00, 1.00) Distribution

Variable	N of Cases	Maximum Difference	Lilliefors Probability (2-tail)
FEMLEN	68	0.099	0.097
FEMWGT	68	0.167	0.000
EGGNO	68	0.224	0.000
BROODVOL	68	0.214	0.000
BROODWGT	68	0.261	0.000
LOGFEMLEN	68	0.095	0.129
LOGFEMWGT	68	0.138	0.003
LOGEGGNO	68	0.103	0.072
LOGBROODVOL	68	0.063	0.695
LOGBROODWGT	68	0.096	0.122

Number of Observations: 68

Pearson Correlation Matrix
	FEMLEN	LOGEGGNO	LOGBROODVOL	LOGBROODWGT
FEMLEN	1.000
LOGEGGNO	0.723	1.000
LOGBROODVOL	0.750	0.904	1.000
LOGBROODWGT	0.760	0.902	0.893	1.000

Matrix of Bonferroni Probabilities
	FEMLEN	LOGEGGNO	LOGBROODVOL	LOGBROODWGT
FEMLEN	0.000
LOGEGGNO	0.000	0.000
LOGBROODVOL	0.000	0.000	0.000
LOGBROODWGT	0.000	0.000	0.000	0.000

Example problems

1. Use the following data on wing length (cm) and tail length (cm) in cowbirds to determine if there is a relationship between the two variables. (Protocol link)

wing 10.4 10.8 11.1 10.2 10.3 10.2 10.7 10.45 10.8 11.2 10.6

tail 7.4 7.6 7.9 7.2 7.4 7.1 7.4 7.2 7.8 7.7 7.8

2. Use the following data taken from crabs to determine if there is a relationship between weight of gills (g) and weight of body (g) and between weight of thoracic shield (g) and weight of body. (Protocol link)

body 159 179 100 45 384 230 100 320 80 220 320

gill 14.4 15.2 11.3 2.5 22.7 14.9 11.4 15.81 4.19 15.39 17.25

thorax 80.5 85.2 49.9 21.1 195.3 111.5 56.6 156.1 39.0 108.9 160.1

________________________________________________________________

Statistical Tests
		Parametric	Nonparametric	Assumptions of parametric tests · Data randomly sampled and independent · Data measured on ratio/interval scale · Variables continuous or discrete (if n and no. possible values large) · Data normally distributed for each group (residuals in ANOVA/regression) · For questions regarding means, data variances among groups (residuals in ANOVA/regression) homogeneous Assumptions of non-parametric tests · Data randomly sampled and independent
Differences	Means (2)	Indep. samples t	Mann-Whitney
		Paired samples t	Wilcoxon
	Means	ANOVA	Kruskal-Wallis
	Variances	Bartlett’s	Levene’s
	Frequencies	-----	Goodness-of-fit
Relationships	Cases	Pearson correlation	Spearman correlation
		Regression	-----
	Frequencies	-----	Independence

Spearman correlation (Chap. 13)

1. Purpose: Test whether the cases of two variables are correlated

2. Comments: if variables are related, the relationship is linear

3. Null hypothesis: H₀: r_{s(var1, var2)} = 0

4. Test statistic (r_s) and probability source: Systat/Statistical Table

5. SYSTAT path: Analyze®Correlation®Simple (enter variables; Rank Order Data)

SYSTAT output: (SHRIMPS.SYD); pic
 
Kolmogorov-Smirnov One Sample Test using Normal(0.00, 1.00) Distribution

Variable	N of Cases	Maximum Difference	Lilliefors Probability (2-tail)
FEMLEN	68	0.099	0.097
FEMWGT	68	0.167	0.000
EGGNO	68	0.224	0.000
BROODVOL	68	0.214	0.000
BROODWGT	68	0.261	0.000
LOGFEMLEN	68	0.095	0.129
LOGFEMWGT	68	0.138	0.003
LOGEGGNO	68	0.103	0.072
LOGBROODVOL	68	0.063	0.695
LOGBROODWGT	68	0.096	0.122

Number of Observations: 68

Spearman Correlation Matrix
	FEMLEN	FEMWGT	EGGNO	BROODVOL	BROODWGT
FEMLEN	1.000
FEMWGT	0.886	1.000
EGGNO	0.725	0.697	1.000
BROODVOL	0.773	0.750	0.905	1.000
BROODWGT	0.749	0.724	0.897	0.884	1.000

Probabilities (not available in SYSTAT, must get from Spearman Table→ P<0.01)

Example problems

1. The following data are ranked scores for ten students who took both a math and a biology aptitude examination. Is there a relationship between math and biology aptitude scores for these students? (Protocol link)

math 53 45 72 78 53 63 86 98 59 71

biology 83 37 41 84 56 85 77 87 70 59

2. Test the following data to determine if there is a relationship between the total length of aphid stem mothers and the mean thorax length of their parthenogenetic offspring. (Protocol link)

mother 8.7 8.5 9.4 10.0 6.3 7.8 11.9 6.5 6.6 10.6

offspring 5.95 5.65 6.00 5.70 4.40 5.53 6.00 4.18 6.15 5.93

____________________________________________________________

Correlation exercise (combination can result from what?)

	Strength of Relationship
	Low (weak)	High (strong)
Significant	?	?
Not significant	?	?

Multiple choice (choose all that apply)

a. biologically important

b. biologically unimportant

c. sufficient power

d. insufficient power

e. high variability

f. other biologically important variables not yet accounted for

________________________________________________________________________________

Statistical Tests
		Parametric	Nonparametric	Assumptions of parametric tests · Data randomly sampled and independent · Data measured on ratio/interval scale · Variables continuous or discrete (if n and no. possible values large) · Data normally distributed for each group (residuals in ANOVA/regression) · For questions regarding means, data variances among groups (residuals in ANOVA/regression) homogeneous Assumptions of non-parametric tests · Data randomly sampled and independent
Differences	Means (2)	Indep. samples t	Mann-Whitney
		Paired samples t	Wilcoxon
	Means	ANOVA	Kruskal-Wallis
	Variances	Bartlett’s	Levene’s
	Frequencies	-----	Goodness-of-fit
Relationships	Cases	Pearson correlation	Spearman correlation
		Regression	-----
	Frequencies	-----	Independence

Independence (Chap. 14)

1. Purpose: Test whether the frequencies of two variables are independent

2. Comments: 2 variables, each frequency occurs in multiple mutually exclusive categories, no proportions or percentages, no cell has an expected frequency of <5 (Systat will inform you of violations)

3. Null hypothesis: H₀: var_(row) independent of var_(column)

4. Test statistic (X²) and probability source: Systat/Systat

5. SYSTAT path: Analyze®Tables®Two-Way (enter row and column variables)

SYSTAT output: (GINMOVE.SYD); SLIDES

Frequencies

HAB$ (rows) by SEX$ (columns)

F M Total

+----------------+

P | 480 420 | 900

R | 2 25 | 27

+----------------+

Total 482 445 927

Test statistic Value DF Prob

Pearson Chi-square 22.1511 1.0000 0.0000

6. Frequency table data - start with table (no raw data)

a. example 1 - association between the hemoglobin S allele and resistance to malaria:

Did not

Contracted contract

malaria malaria

Heterozygotes 1 14

Homozygotes 13 2

SYSTAT output:

Frequencies

MALARIA$ (rows) by GENES$ (columns)

het hom Total

+----------------+

n | 14 2 | 16

y | 1 13 | 14

+----------------+

Total 15 15 30

Test statistic Value DF Prob

Pearson Chi-square 19.286 1.000 0.000

b. example 2 (supplements link)

Example problems

1. Use the following data on frequency of rabies in skunks collected from three geographic areas to test the hypothesis that incidence of rabies dependent on geographic area. (Protocol link)

With Without

Area Rabies Rabies

Ozarks 14 29

Ouachitas 12 38

Delta 11 35

2. The following data are frequency of individuals with different hair colors according to sex. According to these data, is human hair color dependent on sex? (Protocol link)

sex black brown blond red

male 32 43 16 9

female 55 65 64 16

_________________________

Correlation vs. causation

1. spurious correlations; Calvin, Cause and effect
2. sometimes results from a common correlation with 3rd variable (e.g., B correlated with C because both B&C are functionally correlated with A)

Statistical Tests
		Parametric	Nonparametric	Assumptions of parametric tests · Data randomly sampled and independent · Data measured on ratio/interval scale · Variables continuous or discrete (if n and no. possible values large) · Data normally distributed for each group (residuals in ANOVA/regression) · For questions regarding means, data variances among groups (residuals in ANOVA/regression) homogeneous Assumptions of non-parametric tests · Data randomly sampled and independent
Differences	Means (2)	Indep. samples t	Mann-Whitney
		Paired samples t	Wilcoxon
	Means	ANOVA	Kruskal-Wallis
	Variances	Bartlett’s	Levene’s
	Frequencies	-----	Goodness-of-fit
Relationships	Cases	Pearson correlation	Spearman correlation
		Regression	-----
	Frequencies	-----	Independence

Regression (Chap. 12)

1. Purpose: Test whether the cases of one variable are functionally (mathematically) related to the cases of another variable (i.e., can be predicted from)

2. Comments: if variables are related, the relationship is linear; normality assumptions are analyzed with residuals after the regression analysis; robust

3. Null hypothesis: H₀: b_{(vary, varx)} = 0

4. Test statistic (F) and probability source: Systat/Systat

5. SYSTAT path: Analyze®Regression®Linear®Least Squares (enter dependent and independent variables; enter KS on options tab)

Procedure

a. SYSTAT: fit regression line (least squares method; minimize S(residuals²); test for significance of slope

b. Student: determine general regression equation: Y = a + bX (a = intercept; b = slope); parametric, Y = a+ bX

c. Student: determine specific equation (insert regression values and variables)

d. Student: prepare regression plot (use SYSTAT Scatterplot) 

SYSTAT output: (BABIES.SYD; select svl>200 and svl<290 and sex=1); pic

Output format (two tables):

· Regression statistics: intercept (=constant), slope (=regression coefficient); confidence limits (pic)

· ANOVA table

Dep Var: WGT N: 235 Multiple R: 0.874 Squared multiple R: 0.764

Adjusted squared multiple R: 0.763 Standard error of estimate: 0.769

Std Std Toler-

Effect Coefficient Error Coef ance t P(2 Tail)

CONSTANT -16.956 (intercept) 0.991 0.0 . -17.118 0.000

SVL 0.110 (slope) 0.004 0.874 1.000 27.436 0.000

Analysis of Variance

Source Sum-of-Squares DF Mean-Square F-Ratio P

Regression 444.780 1 444.780 752.726 0.000

Residual 137.678 233 0.591

Regression Plot (SYSTAT Scatterplot)

Example problems

1. The following data are rate of oxygen consumption (ml/g/hr) in crows at different temperatures (°C). Does temperature affect oxygen consumption in crows? Determine the equation for predicting oxygen consumption from temperature. (Protocol link)

temp -18 -15 -10 -5 0 5 10 19

oxygen 5.2 4.7 4.5 3.6 3.4 3.1 2.7 1.8

2. Use the following data on mean adult body weight (mg) and larval density (no./mm³) of fruit flies to determine if there is a functional relationship between adult body mass and the density at which it was reared. Determine the equation for predicting body weight from larval density. (Protocol link)

density 1 3 5 6 10 20 40

weight 1.356 1.356 1.284 1.252 0.989 0.664 0.475

Extrapolation: linear regressions are valid only within limits of data (independent variable, X); beyond data - do not know if relationship is linear

 “A regression of tooth size on actual body length for the living Carcharodon carcharias indicates by extrapolation (assuming continued linearity) that C. megalodon was “only” 13 m (43 ft) in length!

 __________________________________________________________

Transformation in linear regression (goal: curvilinear®linear)

1. SYSTAT e.g.: calibrate transmitters; DEMO

2. Linear vs. log10 data regressions - note increase in r² and linearity with log transformation

Dep Var: PI N: 7 Multiple R: 0.989 Squared multiple R: 0.978

Effect Coefficient Std Error Std Coef Tolerance t P

CONSTANT 3172.273 97.857 0.000 . 32.417 0.000

TEMP -65.363 4.390 -0.989 1.000 -14.888 0.000

Analysis of Variance

Source Sum-of-Squares df Mean-Square F-ratio P

Regression 3518606.514 1 3518606.514 221.638 0.000

Residual 79377.200 5 15875.440

Dep Var: LPI N: 7 Multiple R: 1.000 Squared multiple R: 0.999

Effect Coefficient Std Error Std Coef Tolerance t P

CONSTANT 3.540 0.004 0.000 . 834.735 0.000

TEMP -0.015 0.000 -1.000 1.000 -78.645 0.000

Analysis of Variance

Source Sum-of-Squares df Mean-Square F-ratio P

Regression 0.184 1 0.184 6185.068 0.000

Residual 0.000 5 0.000

Predicting dependent variable Y from independent variable X

Linear (Y, X) equations: Y = a + bX

Example 1: using the regression equation Y = 14.5 + 2.56X, predict Y when X = 63

Y = 14.5 + 2.56(63) = 175.78

____________________________________________

Example 2: inverse prediction (predict X from Y); Y = 14.5 + 2.56X

by algebraic manipulation Y-14.5 = 2.56X; (Y-14.5)/2.56 = X

predict X when Y = 175.78:

X = (175.78-14.5)/2.56 = 63

Semilog (logY, X) equations: log Y = log a + bX (must take the inverse log of log Y to get final answer on linear scale)

Example: using the regression equation log Y = 1.42234 +0.47560X, predict Y when X = 12.1

logY = 1.42234 + 0.047560(12.1) = 1.99782 (calculate intercept and

answer to at least 5 decimal places); inverse log 1.99782 = 99.49

Note that the intercept (1.42234) is a log value (i.e., log a = 1.42234). You must not take the log of this value when calculating log Y; that would be the equivalent of taking the log of a log!

Review: logs are exponents

log₁₀a = b is the same as 10^b = a

Log rules

log₁₀(ab) = log₁₀(a) + log₁₀(b)

log₁₀(a/b) = log₁₀(a) – log₁₀(b)

log₁₀(a^b) = blog₁₀(a)

Log-log (logY, logX) equations: log Y = log a + b(log X)

Example 1: using the regression equation log Y = 2.53403 + 0.72000(log X), predict Y when X = 1.98

log Y = 2.53403 + 0.72000 (log 1.98) = 2.74763 (calculate

intercept and answer to at least 5 decimal places)

inverse log 2.74763 = 559.28

An alternative form of the log-log regression equation, and one which is much easier to use is: log Y = log a + b(log X) = log a + log x^b

take inverse logs: Y = aX^b

Example 2: using the regression equation Y = 342X^0.720, predict Y when

X = 1.98; *note that 342 = the inverse log of 2.53403

Y = 342(1.98^0.720) = 559.28

_______________________________________________________________

Examples of important uses of log-log regressions and regression plots in biology

1. ecology: species-area curves (Isle Biogeography Theory)

S = 3.3A^0.30

Common slope in some (~0.3); e.g.,

West Indian snakes: S = 1.19A^0.33

Galapagos land plants: S = 28.6A^0.32

2. physiology: effects of scaling; e.g., metabolic rate and body mass

MR = 70M^0.75

Compare regression lines (ANCOVA); e.g.,

-marsupials: MR = 0.409 M^0.75

-eutherians: MR = 0.676 M^0.75 (>60% higher)

 _____________________________________________________________________________________

Analysis of Covariance (ANCOVA) - not available in MYSTAT

1. Purpose: Detect differences among means of two or more groups when the dependent variable is affected by a third variable (“covariate”); multivariate technique

2. A covariate is a continuous independent variable that adds unwanted variability to the dependent variable

3. ANCOVA removes the variability in the dependent variable due to the covariate

4. ANCOVA combines the use of both ANOVA and regression methods

5. ANCOVA permits accessing interaction between independent variables

6. Assumptions (normality and equal variances of residuals) are checked after ANCOVA

7. SYSTAT path: Analyze®General Linear Model®Estimate Model® enter dependent and independent variables (categorical, covariate); enter interaction terms; enter KS and Levene on options tab
8. MYSTAT: not available

Example: (pic) Does residual yolk mass differ between fed and unfed turtle hatchlings? (pdf)

1. One approach is to measure yolk mass on hatchlings and analyze mass data with a t-test or ANOVA.

Analysis of Variance (ANOVA)
Source	SS	df	Mean Square	F-ratio	p-value
Treat	0.022	1	0.022	0.063	0.803 ns
Error	28.325	82	0.345

Note the two sources of variation and the

relatively large (94%) error variance.

2. Confounding factor: yolk mass is a function of age (r²~80%); incorporate age variation into the analysis (ANCOVA)

Analysis of Variance (ANCOVA)
Source	SS	df	Mean Square	F-ratio	p-value
Treat	0.002	1	0.002	0.030	0.862 ns
Age	22.608	1	22.608	318.561	0.000 ***
Treat*Age	0.007	1	0.007	0.099	0.754
Error	5.678	80	0.071

Note the additional sources of variation and

greatly reduced (<1%) error variance.

3. Student responsibility for ANCOVA

- have basic understanding

- be able to interpret output

- be able to work and explain the example problem (no protocol sheet)

ANCOVA example problem

1. A common belief is that men are stronger than women. Is this belief due to men being bigger or are men actually stronger when compared to women of similar body size? Test this question on data from a sample of healthy young adults (stronger.syd). The variables are sex, lean body mass, and a measure of strength called “slow, right extensor knee peak torque.”

Multivariate statistics (univariate, bivariate, multivariate)

· Examples: Two-way ANOVA, ANCOVA

· Other multivariate procedures commonly used in the literature

a. principal components

b. MANOVA

c. factor analysis

d. logistic regression

e. multiple regression

A. Multiple regression example (>1 independent variable); DEMO FYI only

1. Stepwise method (Analyze®Regression®Linear®Least Squares®Options®Stepwise)

- constructs model with the highest overall r²

- adds variables in steps according to the strength of a significant relationship

- graph and equation of model