Background
Introduction
Parametric test are those statistical approaches which rely on assumptions
about the parameters which define a population.
Prominent parametric tests include:
Pearson correlation (Seminar 9 - Correlation Tests)
t-Test
Analysis Of Variance (ANOVA)
Linear regression
Multivariate extensions of parametric methods
...
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Background
Terminology
A reminder about the distinction of parametric and non-parametric tests (taken
from Seminar 6):
Non-Parametric Tests
Less restrictive
Make little to no assumptions
Often a black box
Require more data
Parametric Tests
More restrictive
Make strict assumptions
Easy to interpret
Require less data
→ Parametric tests are numerous!
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Analyses t-Test (unpaired)
Purpose And Assumptions
t-Test (unpaired)
t.test(..., paired = FALSE) in base R
Pur pose:
To identify whether groups of variable values are different from one
another.
H
0
There is no difference in characteristics of the response variable
values in dependence of the classes of the predictor variable.
Assumptions:
Predictor variable is binary
Response variable is metric and normal distributed within their
groups
Variable values are independent (not paired)
→ Test whether variance of response variable values in groups are equal (var.test()) and adjust t.test()
argument var.equal accordingly.
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Analyses t-Test (unpaired)
Minimal Working Example
Let’s feed data to our t.test(..., paired = FALSE) function that holds two
groups with clearly differing means:
data <- c(rnorm(10, 5, 1), rnorm(10, 10, 1))
factors <- as.factor(rep(c("A", "B"), each = 10))
t.test(data ~ factors, paired = FALSE)
##
## Welch Two Sample t-test
##
## data: data by factors
## t = -12, df = 14, p-value = 1e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -6.4 -4.4
## sample estimates:
## mean in group A mean in group B
## 4.4 9.8
The output above tells us that the means of our two groups are significantly different.
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Analyses t-Test (paired)
Purpose And Assumptions
t-Test (paired)
t.test(..., paired = TRUE) in base R
Pur pose:
To identify whether groups of variable values are different from one
another.
H
0
There is no difference in characteristics of the response variable
values in dependence of the classes of the predictor variable.
Assumptions:
Predictor variable is binary
Response variable is metric
Difference of response variable pairs is normal distributed
Variable values are dependent (paired)
→ Test whether variance of response variable values in groups are equal (var.test()) and adjust t.test()
argument var.equal accordingly.
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Analyses t-Test (paired)
Minimal Working Example
Let’s feed data to our t.test(..., paired = TRUE) function that holds two
connected groups with clearly differing means:
data <- c(rnorm(10, 5, 1), rnorm(10, 10, 1))
factors <- as.factor(rep(c("A", "B"), each = 10))
t.test(data ~ factors, paired = TRUE)
##
## Paired t-test
##
## data: data by factors
## t = -10, df = 9, p-value = 3e-06
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -5.7 -3.7
## sample estimates:
## mean of the differences
## -4.7
The output above tells us that the means of our two connected groups are significantly
different.
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Analyses Analysis of Variance (ANOVA)
Introduction to ANOVA
ANOVAs are used to test whether there is a difference between groups of
variable values.
There are multiple versions of ANOVAs:
One-way ANOVA (one predictor variable)
Two-Way ANOVA (multiple predictor variables)
MANOVA (multivariate ANOVA/multiple response variables)
ANCOVA (categorical and continuous predictor variables)
MANCOVA (multivariate ANCOVA)
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Analyses Analysis of Variance (ANOVA)
Data for ANOVA
We will use the crabs data set from the MASS package
library(MASS)
data(crabs)
head(crabs)
## sp sex index FL RW CL CW BD
## 1 B M 1 8.1 6.7 16 19 7.0
## 2 B M 2 8.8 7.7 18 21 7.4
## 3 B M 3 9.2 7.8 19 22 7.7
## 4 B M 4 9.6 7.9 20 23 8.2
## 5 B M 5 9.8 8.0 20 23 8.2
## 6 B M 6 10.8 9.0 23 26 9.8
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Analyses One-Way ANOVA
Purpose And Assumptions
One-Way ANOVA
anova() in base R
Pur pose:
To explain the variance of a continuous response variable in relation to
one predictor variables.
H
0
Variance of response variable values is equal between levels of
predictor variable.
Assumptions:
Predictor variable is categorical
Response variable is metric
Response variable residuals are normal distributed
Variance of populations/samples are equal (homogeneity)
Variable values are independent (not paired)
→ Test whether residuals are normal distributed with shapiro.test() in base R, test for homogeneity with
leveneTest() in the car package.
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Analyses One-Way ANOVA
Minimal Working Example - Assumptions
Let’s test whether body depth (BD) of crabs are varying when grouped by sex:
OneWay <- with(crabs, lm(BD ~ sex)) # MODEL
plot(OneWay, 2)# Normality
−3 −2 −1 0 1 2 3
−2 −1 0 1 2
Theoretical Quantiles
Standardized residuals
lm(BD ~ sex)
Normal Q−Q
51
200
1
shapiro.test(residuals(OneWay))
##
## Shapiro-Wilk normality test
##
## data: residuals(OneWay)
## W = 1, p-value = 0.2
plot(OneWay, 3)# Homogeneity
13.7 13.8 13.9 14.0 14.1 14.2 14.3
0.0 0.5 1.0 1.5
Fitted values
Standardized residuals
lm(BD ~ sex)
Scale−Location
51
200
1
library("car")
leveneTest(BD ~ sex, data = crabs)
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 1 0.36 0.55
## 198
All good on the assumption check!
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Analyses One-Way ANOVA
Minimal Working Example - Analysis
Now let’s run the analysis:
anova(OneWay)
## Analysis of Variance Table
##
## Response: BD
## Df Sum Sq Mean Sq F value Pr(>F)
## sex 1 19 18.8 1.61 0.21
## Residuals 198 2315 11.7
As we can see, sex does not make for a statistically significant predictor of crab
body depth.
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Analyses One-Way ANOVA
Minimal Working Example - Interpretation
Let’s interpret the result anyways:
summary(OneWay)
##
## Call:
## lm(formula = BD ~ sex)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.624 -2.449 0.076 2.463 7.376
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.724 0.342 40.13 <2e-16
***
## sexM 0.613 0.484 1.27 0.21
## ---
## Signif. codes:
## 0 '
***
' 0.001 '
**
' 0.01 '
*
' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.4 on 198 degrees of freedom
## Multiple R-squared: 0.00805, Adjusted R-squared: 0.00304
## F-statistic: 1.61 on 1 and 198 DF, p-value: 0.206
Female crabs are estimated to have a body depth of 13.72cm (Intercept) with males being 0.61cm
bigger, on average.
While we can be certain of the female estimate, we cannot say the same about the different to males.
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Analyses Two-Way ANOVA
Purpose And Assumptions
Two-Way ANOVA
anova() in base R
Pur pose:
To explain the variance of a continuous response variable in relation to
multiple predictor variables.
H
0
Variance of response variable values is equal between levels of
predictor variables.
Assumptions:
Predictor variables are categorical
Response variable is metric
Response variable residuals are normal distributed
Variance of populations/samples are equal (homogeneity)
Variable values are independent (not paired)
→ Test whether residuals are normal distributed with shapiro.test() in base R, test for homogeneity with
leveneTest() in the car package.
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Analyses Two-Way ANOVA
Minimal Working Example - Assumptions
Let’s test whether body depth (BD) of crabs are varying when grouped by sex
and species as well as their interaction:
TwoWay <- with(crabs, lm(BD ~ sex
*
sp))
plot(TwoWay, 2)# Normality
−3 −2 −1 0 1 2 3
−2 −1 0 1 2
Theoretical Quantiles
Standardized residuals
lm(BD ~ sex * sp)
Normal Q−Q
101
50
1
shapiro.test(residuals(TwoWay))
##
## Shapiro-Wilk normality test
##
## data: residuals(TwoWay)
## W = 1, p-value = 0.2
plot(TwoWay, 3)# Homogeneity
12 13 14 15
0.0 0.5 1.0 1.5
Fitted values
Standardized residuals
lm(BD ~ sex * sp)
Scale−Location
101
50
1
library("car")
leveneTest(BD ~ sex
*
sp, data = crabs)
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 3 2.02 0.11
## 196
All good on the assumption check!
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Analyses Two-Way ANOVA
Minimal Working Example - Analysis
Now let’s run the analysis:
anova(TwoWay)
## Analysis of Variance Table
##
## Response: BD
## Df Sum Sq Mean Sq F value Pr(>F)
## sex 1 19 19 1.99 0.160
## sp 1 419 419 44.31 2.8e-10
***
## sex:sp 1 42 42 4.48 0.035
*
## Residuals 196 1854 9
## ---
## Signif. codes:
## 0 '
***
' 0.001 '
**
' 0.01 '
*
' 0.05 '.' 0.1 ' ' 1
The output above tells us that species and the interaction effect of sex and
species are meaningful for understanding body depth of crabs.
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Analyses Two-Way ANOVA
Minimal Working Example - Interpretation
summary(TwoWay)
##
## Call:
## lm(formula = BD ~ sex
*
sp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.924 -2.224 0.059 2.250 6.650
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.816 0.435 27.17 < 2e-16
***
## sexM 1.534 0.615 2.49 0.013
*
## spO 3.816 0.615 6.20 3.2e-09
***
## sexM:spO -1.842 0.870 -2.12 0.035
*
## ---
## Signif. codes:
## 0 '
***
' 0.001 '
**
' 0.01 '
*
' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.1 on 196 degrees of freedom
## Multiple R-squared: 0.206, Adjusted R-squared: 0.194
## F-statistic: 16.9 on 3 and 196 DF, p-value: 8.13e-10
Female crabs of species B are estimated to have a body depth of 11.82cm (Intercept) with males of species B being 1.53cm bigger, on average.
Female crabs of species O are estimated to have a body depth of 3.82cm bigger than their female species B counterparts.
The difference in sex- vs. species-dependant change in body depth is -1.84cm.
All estimates are statistically significant.
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Analyses ANCOVA
Purpose And Assumptions
ANCOVA
anova() in base R
Pur pose:
To explain the variance of a continuous response variable in relation to
mixed (continuous and categorical) predictor variables.
H
0
Adjusted variance and means of response variable values is equal
between levels of predictor variables.
Assumptions:
Predictor variables are categorical or continuous
Response variable is metric
Response variable residuals are normal distributed
Variance of populations/samples are equal (homogeneity)
Variable values are independent (not paired)
Relationship between the response and covariate is linear.
→ Test whether residuals are normal distributed with shapiro.test() in base R, test for homogeneity with leveneTest() in the car package.
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Analyses ANCOVA
Minimal Working Example - Assumptions
Let’s test whether carapace length (CL) of crabs are varying when grouped by
species and the carapace width as a covariate:
Ancova <- with(crabs, lm(CL ~ sp
*
CW))
plot(Ancova, 2)# Normality
−3 −2 −1 0 1 2 3
−3 −2 −1 0 1 2 3
Theoretical Quantiles
Standardized residuals
lm(CL ~ sp * CW)
Normal Q−Q
188
70
145
shapiro.test(residuals(Ancova))
##
## Shapiro-Wilk normality test
##
## data: residuals(Ancova)
## W = 1, p-value = 0.2
plot(Ancova, 3)# Homogeneity
15 20 25 30 35 40 45
0.0 0.5 1.0 1.5
Fitted values
Standardized residuals
lm(CL ~ sp * CW)
Scale−Location
188
70
145
library("car")
leveneTest(CL ~ sp, data = crabs)
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 1 0.1 0.75
## 198
Assumptions are met!
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Analyses ANCOVA
Minimal Working Example - Analysis
Now let’s run the analysis:
anova(Ancova)
## Analysis of Variance Table
##
## Response: CL
## Df Sum Sq Mean Sq F value Pr(>F)
## sp 1 838 838 3868.20 <2e-16
***
## CW 1 9203 9203 42460.12 <2e-16
***
## sp:CW 1 1 1 4.29 0.04
*
## Residuals 196 42 0
## ---
## Signif. codes:
## 0 '
***
' 0.001 '
**
' 0.01 '
*
' 0.05 '.' 0.1 ' ' 1
The output above tells us that all of our model coefficients are significant.
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Analyses ANCOVA
Minimal Working Example - Interpretation
summary(Ancova)
##
## Call:
## lm(formula = CL ~ sp
*
CW)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4634 -0.2611 -0.0041 0.2907 1.1861
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.36442 0.21170 -1.72 0.087 .
## spO 0.44111 0.32079 1.38 0.171
## CW 0.87630 0.00595 147.31 <2e-16
***
## spO:CW 0.01781 0.00860 2.07 0.040
*
## ---
## Signif. codes:
## 0 '
***
' 0.001 '
**
' 0.01 '
*
' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.47 on 196 degrees of freedom
## Multiple R-squared: 0.996, Adjusted R-squared: 0.996
## F-statistic: 1.54e+04 on 3 and 196 DF, p-value: <2e-16
Crabs of species B have an estimated carapace length of -0.36cm when their carpace width would be 0cm (Intercept) with members of species B
being 0.44cm bigger, on average at 0cm carapace width.
For each additional cm in carapace width, carapacae length in species B increases by 0.88cm.
For each additional cm in carapace width, carapacae length in species O increases by 0.88cm more than in species B.
All estimates except for the species-difference are statistically significant.
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Our Data Choice Of Variables
Variables We Can Use
Response variables (metric)
Weight
Height
Wing Chord
Nesting Height
Number of Eggs
Egg Weight
Predictor variables (categorical)
Sex (binary)
Climate (binary)
Climate (3 levels - Continental,
Semi-Coastal, Coastal)
Home Range (3 levels - Small,
Medium, Large)
Site Index (11 levels)
Predator Presence/Type (3 levels -
Avian vs. Non-Avian vs. None)
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Our Data Research Questions
Research Questions And Hypotheses
So which of our major research questions (seminar 6) can we answer?
unpaired t-Test
Climate Warming/Extremes: Does sparrow
morphology change depend on climate?
Sexual Dimorphism: Does sparrow morphology
change depend on Sex?
Use the 1 - Sparrow_Data_READY.rds data set for these analyses.
paired t-Test (suppose a resettling program)
Climate Warming/Extremes: Does sparrow
morphology change depend on climate?
Use the 2b - Sparrow_ResettledSIUK_READY.rds data set for these
analyses.
One-Way ANOVA
Climate Warming/Extremes: Does sparrow
morphology depend on climate?
Predation: Does nesting height depend on
predator characteristics?
Two-Way ANOVA
Sexual Dimorphism: Does sparrow morphology
depend on population status and sex?
ANCOVA
Climate Warming/Extremes: Do sparrow
characteristics depend on climate and latitude?
Use the 1 - Sparrow_Data_READY.rds data set for these analyses.
Remember to diligently check assumptions!
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