Transcript Presentation 2

Research Design
What is the Question we want to Answer?
What are the relevant Variables?
What magnitude of effect do we expect to observe if our
hypothesis is true?
How much variation do we expect to see in our data
that might obscure this effect?
How many samples do we have to gather in order to
observe an effect of a specified magnitude?
Hypothesis Formation
Underlying Causes to Explain our Observations
Proximate

Causes due to short term responses to some perturbation
Ultimate

The historical reason that some phenomenon came into
existence in the first place
Hypotheses
Null Hypothesis
The outcome you expect if the hypothesized
underlying cause is not driving the observed
phenomenon
Alternative Hypothesis
The outcome you expect if the hypothesized
underlying cause is driving the observed
phenomenon
Statistical Analyses
Which analyses are appropriate for your data?
What is the nature of your data?
Nominal (Discrete)
Ordinal
Continuous
What transformations of your data will be
required?
Interpretation (Conclusion)
What is the answer to the original question?
Constrained by your data
To what degree did your experiment allow you to
answer the question?
What portions of the question are unresolved?
What new questions have been generated by the
answer you got?
How does your answer relate to the answers others
have found?
Analysis of Variance
One-way analysis of variance
Data Issues
Types of Data
Overlay
PlotDistribution
Normal
Normal Density
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
St. Dev.
1
2
3
4
Non-Normal Distributions
Bimodal
Symmetrical
Non-Normal
Not symmetrical
Overlay
PlotDistribution
Normal
Normal Density
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
St. Dev.
1
2
3
4
Overlay
PlotDistribution
Normal
Normal Density
0.4
0.3
0.2
0.1
0
2.5%
-4
-3
-2
95%
-1
0
St. Dev.
2.5%
1
2
3
4
Overlay
PlotDistribution
Normal
Normal Density
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
St. Dev.
1
2
3
4
Overlay
PlotDistribution
Normal
x
Normal Density
0.4
0.3
x  xij
0.2
x  xij
0.1
0
-4
-3
-2
-1
0
St. Dev.
1
2
3
4
Variance, Standard Deviation and
Standard Error
s s

 n
  x  xij

j 1

2
s 
n  1

2
2




S .E . 
s
n
Analysis of Variance (ANOVA)
Frequency
x1  x 2
1.96(S.E.)
Low
1.96(S.E.)
Variable of Interest
High
Analysis of Variance (ANOVA)
Frequency
x1  x 2
1.96(S.E.)
Low
1.96(S.E.)
Variable of Interest
High
Analysis of Variance (ANOVA)
Frequency
x1  x 2
Low
Variable of Interest
High
Analysis of Variance (ANOVA)
Frequency
x1  x 2
Low
Variable of Interest
High
Frequency
Analysis of Variance (ANOVA)
Low
Variable of Interest
High
Assumptions of Analysis of Variance
Data from the samples are independent
Data are normally distributed (or can be
transformed to be so)
The variances of the different samples are not
heterogeneous
For each individual you must have score
in two variables:
Score A – dependent variable - quantitative
Score B – factor – Divide individuals between
2 or more groups or level
ANOVA F test evaluates whether the group means on the
Dependent variable
Example: Experimental Study – Lesson 24
A sample of 30 volunteers
factor
placebo
Low doses
high doses
Number of days that they have cold symptoms
dependent
Effect size Statistic
Eta2 ranges in value from 0 to 1.
An eta2= 0 indicates that there are No differences in the
mean scores among groups
Eta2= 1 indicates that there are differences between at
least two of the means on the dependent variable
effect sizes of eta2=0.01 (small)
eta2=0.06 (medium)
eta2=0.14 (large)
Research Question
Mean Differences
Relationship between variables
ANOVA Table
Source of
Variation
Treatments
Degrees of
freedom (df)
Sum of Squares
g-1
g
SStr   nl  xl  x 2
l 1
Residual
(error)
Total
(corrected for the
mean)
SSres    nl xlj  xl 
g nl
2
l 1 j 1
l 1 j 1
 nl  g
l 1
SScor    nl xlj  x 
g nl
g
2
g
 nl  1
l 1
Descriptive Statistics
Dependent Variable: DIFF
Vitamin C Treatment
1
2
3
Total
Mean
3.50
-2.10
-2.00
-.20
Std. Deviation
4.143
4.067
5.477
5.182
N
10
10
10
30
variances
from 16,
to 25)
Levene's Test of Equality of Error Variancesa
Dependent Variable: DIFF
F
1.343
df1
df2
2
27
Sig.
.278
Tes ts the null hypothes is that the error variance of
the dependent variable is equal across groups.
a. Des ign: Intercept+GROUP
Variance is not homeg.
Large Effect size
Tests of Between-Subjects Effects
Dependent Variable: DIFF
Source
Corrected Model
Intercept
GROUP
Error
Total
Corrected Total
Type III Sum
of Squares
205.400 a
1.200
205.400
573.400
780.000
778.800
df
2
1
2
27
30
29
Mean Square
102.700
1.200
102.700
21.237
F
4.836
.057
4.836
Sig.
.016
.814
.016
Partial Eta
Squared
.264
.002
.264
a. R Squared = .264 (Adjus ted R Squared = .209)
P=0.016 (significant)
Large effect size: strong relationship between vitamin C factor and
cold days
Multiple Comparisons
Dependent Variable: DIFF
Tukey HSD
(I) Vitamin C Treatment
1
2
3
Dunnett T3
1
2
3
(J) Vitamin C Treatment
2
3
1
3
1
2
2
3
1
3
1
2
Bas ed on observed means.
*. The mean difference is significant at the .05 level.
Mean
Difference
(I-J)
5.60*
5.50*
-5.60*
-.10
-5.50*
.10
5.60*
5.50
-5.60*
-.10
-5.50
.10
Std. Error
2.061
2.061
2.061
2.061
2.061
2.061
1.836
2.172
1.836
2.157
2.172
2.157
Sig.
.030
.033
.030
.999
.033
.999
.020
.062
.020
1.000
.062
1.000
95% Confidence Interval
Lower Bound Upper Bound
.49
10.71
.39
10.61
-10.71
-.49
-5.21
5.01
-10.61
-.39
-5.01
5.21
.79
10.41
-.23
11.23
-10.41
-.79
-5.79
5.59
-11.23
.23
-5.59
5.79
20
1
10
3
0
DIFF
-10
-20
N=
10
10
10
1
2
3
Vitamin C Treatment
Sum of Squares (SS)
Total variation =
Variation due to patients
+ Variation due to drugs
+ Unexplained variation
Variation explained by model
SS (total) = SS (model) + SS (residual) =
SS (patients) + SS (drugs) + SSE