[Middle Term]
BIO EPI – Biostatistics and Epidemiology
Quantitative Response Variable
Inferences about the Mean:
• estimate of precision between sample mean and population mean
• Standard Error of the Mean (Semean)
STEPS IN MAKING A VALID INFERENCE
STEP 1 (Sampling)
STEP 2 (Analysis)
Quantitative Response
Variable derived from:
• Single Simple Random
Var.
• Match-Pair Sample
• Describe & explore the
data
• Summary Statistics
• Graphical Techniques
COMPARING TWO MEANS
• simplest study design
• determines the effect of Independent (X)
to Dependent (Y) [msrd. in 2 groups]
STEP 3
• Hypothesis Testing
DATA REQUIREMENTS
• Simple random sample
• Normal Source N (or a large n*)
• Valid data measurement
TEST FOR NORMALITY
t-PROCEDURES
Comparison of:
Concurrent
Control Group:
Problems:
One Sample
• Results to Previously Est.
Norms and Expected
Values
• One sample mean to a
null hypothesis value
None
• Sample from a single
group
• One numerical or
ordinal variable of
interest
Examples:
Paired Sample
• Two population means
• Before and After
• Difference between
paired observations
None
Independent
• Response between 2
groups
• Unrelated data points
between groups
• Equal or unequal s2
Yes
• Two samples that are
related to each other
(dependence between
two samples)
• One numerical or
ordinal variable of
interest
• Two independent
(unrelated) groups of
individuals
• One numerical or
ordinal variable of
interest
• Pretest/Post-test
(1 treatment on one
sample)
• Cross over trial
(2 treatments on one
sample)
• Experimental vs.
Control Group
Assumption:
• Normally distributed
variable
• Reasonable sample size
to check assumption of
Normality
• Normally distributed
variable
• Reasonable sample size
to check assumption of
Normality
• Normally distributed
variable in each group
• Variances are the
same (usually
assumed)
• Reasonable sample
size to check
assumption of
Normality and equal
variances
If Assumption is
not Satisfied:
• Transform the data so
that the variable is
Normally dist.
• Use nonparametric test
▪ Sign test
▪ Wilcoxon Signed
Rank test
• Transform the data so
that the variable is
Normally distributed
• Use nonparametric test
▪ Sign test
▪ Wilcoxon signed
rank test
• Transform the data so
that the variable is
Normally distributed
• Use t test result
assuming unequal
variances
• Use nonparametric
test
▪ Wilcoxon Rank
Sum Test
▪ Mann Whitney U
Test
ONE SAMPLE t-TEST
PAIRED SAMPLE t-TEST
Comparing Several Means (ANOVA)
Analysis of Variance (ANOVA)
• Determines variations within each group
• Allows insight into group differences
•
•
•
•
•
CONCERNS
Problem of multiplicity
Problem of multiple
comparisons
More comparison →
higher Type I Error rate
PROBLEMS
Sample from a number
of independent groups
One numerical or
ordinal variable of
interest
•
•
•
•
•
•
PREVENTION
Determine overall
significance before
comparing two groups at
a time
Allows identifying specific
areas of differences
ASSUMPTIONS
Sample from a number of
independent groups
Normally dist. variable
Equal variances of groups
Reasonable sample size to
check assumption of
Normality and equal s2
SCEDASTICITY
• Variance of a random variable or
the degree or variation
•
•
•
•
•
REQUIREMENTS
Sampling independence
within groups
Normality of sample mean
distribution
Equal variance in the
source population
If Not Satisfied:
Transform the data so that
the variable is Normally
distributed
Use nonparametric test
• Kruskal Wallis test
Types of Scedasticity
1. Homoscedastic (Equal Variances)
2. Heteroscedastic (Unequal Variances)
Graphical Exploration
• Discrepant IQRs = HTSC
ASSESSMENT OF SCEDASTICITY
Summary Statistics
• If the SD (one group) > 2x
•
SD (other group) = HTSC
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•
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TYPES OF ANOVA TEST
One-Way ANOVA
Comparison
of:
•
•
•
•
Means of 2 or more groups
1 Independent Variable
1 Dependent Variable
Divides the variance in a data set
o s2 Between Groups
▪ MS Between
▪ Quantifies the variance
between group means
around the grand mean
▪ Sum of squares between
groups divided by the
degrees of freedom
between groups
2
o s Within Groups
▪ MS Within
▪ Quantifying variability of
data points in a group
around its mean
▪ Sum of squares within
groups divided by the
degrees of freedom within
groups
• F-STATISTIC
o Ratio of variances between
and within groups
o “signal to noise” ratio
▪ Signal: group difference
▪ Noise: variation within
groups
o AIM: larger F ratio
Hypothesis Testing
F ratio test
Bartlett’s test
Levene’s Test
Usually included in ANOVA
procedure
Two-Way ANOVA
• Means of 2 or more groups
• 2 Independent Variable
• 1 Dependent Variable
If H0 is Rejected:
• At least one of the population
mean differs
• Does not indicate which one
• Requires Post-Hoc Comparisons
POST HOC COMPARISONS
1. LSD: Least Square Difference
2. Bonferroni Method
3. Scheffe Method
4. Tukey Method (most robust)
5. Duncan Method
For Unequal Variances:
• Use descriptive analyses instead
• Address outliers
• Use robust methods
ADDRESSING OUTLIERS
• Determine the cause of outliers
• If caused by errors in data →
corrected
• If errors cannot be corrected,
remove outliers
• If from another population,
analyze
• If from within the population,
use non-normal distribution
methods
USING ROBUST METHODS
• Use an inferential technique that is
still valid under conditions of nonNormality or unequal variance
• Bootstrap methods, permutation
tests, distribution free methods
and nonparametric tests
NONPARAMETRIC RANK-BASED
TESTS
• Replacing values in the data set
with their associated ranks
• Focus on relation positions within
ordered data