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,
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)
|
|
Parametric |
Nonparametric |
·
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 |
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|
Differences |
Means (2) |
Indep. samples t |
Mann-Whitney |
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Paired samples t |
Wilcoxon |
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Means |
ANOVA |
Kruskal-Wallis |
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Variances |
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Levene’s |
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Frequencies |
----- |
Goodness-of-fit |
|||
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Relationships |
Cases |
Pearson correlation |
Spearman
correlation |
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Regression |
----- |
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Frequencies |
----- |
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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)
Variable N of Cases Maximum Lilliefors 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 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
SYSTAT output: (SHRIMPS.SYD); pic
Kolmogorov-Smirnov One Sample Test using Normal(0.00, 1.00) Distribution
Difference
Probability
(2-tail)
Number of Observations: 68
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
________________________________________________________________
|
|
Parametric |
Nonparametric |
·
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 |
|
Levene’s |
||
|
Frequencies |
----- |
Goodness-of-fit |
||
|
Relationships |
Cases |
Pearson
correlation |
Spearman
correlation
|
|
|
Regression |
----- |
|||
|
Frequencies |
----- |
|
||
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: rs(var1, var2) = 0
4. Test statistic (rs) and probability
source: Systat/Statistical Table
5. SYSTAT path: Analyze®Correlation®Simple (enter
variables; Rank Order Data)
Variable N of Cases Maximum Lilliefors 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)
SYSTAT output: (SHRIMPS.SYD); pic
Kolmogorov-Smirnov One Sample Test using Normal(0.00, 1.00) Distribution
Difference
Probability
(2-tail)
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?)
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Strength of Relationship |
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Low (weak) |
High (strong) |
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Significant |
? |
? |
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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
________________________________________________________________________________
|
|
Parametric |
Nonparametric |
·
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 |
|
Levene’s |
||
|
Frequencies |
----- |
Goodness-of-fit |
||
|
Relationships |
Cases |
Pearson
correlation |
Spearman
correlation |
|
|
Regression |
----- |
|||
|
Frequencies |
----- |
|
||
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)
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
|
|
Parametric |
Nonparametric |
·
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 |
|
Levene’s |
||
|
Frequencies |
----- |
Goodness-of-fit |
||
|
Relationships |
Cases |
Pearson
correlation |
Spearman
correlation |
|
Regression
|
----- |
|||
|
Frequencies |
----- |
|
||
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
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
density 1 3 5 6 10 20 40
weight 1.356 1.356 1.284 1.252 0.989 0.664 0.475

“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 10b = a Log rules log10(ab) = log10(a) + log10(b) log10(a/b) = log10(a) – log10(b) log10(ab) =
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 xb take inverse logs:
Y = aXb Example 2:
using the regression equation Y = 342X0.720, predict Y
when X = 1.98; *note that 342 = the inverse log of 2.53403 Y = 342(1.980.720) = 559.28
_______________________________________________________________
S
= 3.3A0.30

Common
slope in some (~0.3); e.g., West
Indian snakes: S = 1.19A0.33 Galapagos
land plants: S = 28.6A0.32
MR = 70M0.75

Compare regression lines (ANCOVA); e.g., -marsupials: MR =
0.409 M0.75 -eutherians: MR =
0.676 M0.75 (>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
Example:
(pic)
Does residual yolk mass differ between fed and unfed turtle hatchlings? (pdf)

|
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
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
