statistical analysis of the two group post-only randomized experiment
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Statistical Analysis of the Two Group Statistical Analysis of the Two Group Post-Only Randomized ExperimentPost-Only Randomized Experiment
Analysis RequirementsAnalysis Requirements
Two groupsTwo groups A post-only measureA post-only measure Two distributions, each with an average and variationTwo distributions, each with an average and variation Want to assess treatment effectWant to assess treatment effect Treatment effect = statistical (i.e., nonchance) Treatment effect = statistical (i.e., nonchance)
difference between the groupsdifference between the groups
R X OR O
Statistical AnalysisStatistical Analysis
Statistical AnalysisStatistical Analysis
Controlgroupmean
Statistical AnalysisStatistical Analysis
Controlgroupmean
Treatmentgroupmean
Statistical AnalysisStatistical Analysis
Controlgroupmean
Treatmentgroupmean
Is there a difference?
What Does What Does DifferenceDifference Mean? Mean?
What Does What Does DifferenceDifference Mean? Mean?
Mediumvariability
What Does What Does DifferenceDifference Mean? Mean?
Mediumvariability
Highvariability
What Does What Does DifferenceDifference Mean? Mean?
Mediumvariability
Highvariability
Lowvariability
What Does What Does DifferenceDifference Mean? Mean?
Mediumvariability
Highvariability
Lowvariability
The mean differenceis the same for all
three cases.
What Does What Does DifferenceDifference Mean? Mean?
Mediumvariability
Highvariability
Lowvariability
Which one showsthe greatestdifference?
What Does What Does DifferenceDifference Mean? Mean? A statistical difference is a function of the A statistical difference is a function of the
difference between meansdifference between means relative to the relative to the variabilityvariability..
A small difference between means with large A small difference between means with large variability could be due to variability could be due to chancechance..
Like a Like a signal-to-noisesignal-to-noise ratio. ratio.
Lowvariability
Which one showsthe greatestdifference?
What Do We Estimate?What Do We Estimate?
Lowvariability
What Do We Estimate?What Do We Estimate?
Lowvariability
Signal
Noise
What Do We Estimate?What Do We Estimate?
Lowvariability
Signal
Noise
Difference between group means=
What Do We Estimate?What Do We Estimate?
Lowvariability
Signal
Noise
Difference between group means
Variability of groups=
What Do We Estimate?What Do We Estimate?
Lowvariability
Signal
Noise
Difference between group means
Variability of groups=
=XT - XC
SE(XT - XC)
_ _
_ _
What Do We Estimate?What Do We Estimate?
Lowvariability
Signal
Noise
Difference between group means
Variability of groups=
XT - XC
SE(XT - XC)=
= t-value
_ _
_ _
What Do We Estimate?What Do We Estimate?
The t-test, one-way analysis of variance The t-test, one-way analysis of variance (ANOVA) and a form of regression all test the (ANOVA) and a form of regression all test the same thing and can be considered equivalent same thing and can be considered equivalent alternative analyses.alternative analyses.
The regression model is emphasized here The regression model is emphasized here because it is the most general.because it is the most general.
Lowvariability
Regression Model for t-Test or One-Way Regression Model for t-Test or One-Way ANOVAANOVA
yi = 0 + 1Zi + ei
Regression Model for t-Test or One-Way Regression Model for t-Test or One-Way ANOVAANOVA
yyii = = outcome score for the ioutcome score for the ithth unit unit
00 == coefficient for the coefficient for the interceptintercept
11 == coefficient for the coefficient for the slopeslope
ZZii == 1 if i1 if ithth unit is in the treatment group unit is in the treatment group
0 if i0 if ithth unit is in the control group unit is in the control groupeeii == residual for the iresidual for the ithth unit unit
yi = 0 + 1Zi + ei
where:
In Graph Form...In Graph Form...
In Graph Form...In Graph Form...
0(Control)
1(Treatment) Zi
In Graph Form...In Graph Form...
0(Control)
1(Treatment)
Yi
Zi
In Graph Form...In Graph Form...
0(Control)
1(Treatment)
Yi
Zi
In Graph Form...In Graph Form...
0(Control)
1(Treatment)
0 is the intercepty-value when z=0.
Yi
Zi
In Graph Form...In Graph Form...
0(Control)
1(Treatment)
0 is the intercepty-value when z=0.
1 is the slope.
Yi
Zi
Why Is 1 the Mean Difference?
0(Control)
1(Treatment)
0 is the intercepty-value when z=0.
1 is the slope.
Yi
Zi
Why Is 1 the Mean Difference?
0(Control)
1(Treatment)
Intuitive Explanation:Because slope is the change in y for a 1-unit change in x.
Yi
Zi
Change in y
Unit change in x (i.e., z)
Why Is 1 the Mean Difference?
0(Control)
1(Treatment)
Since the 1-unit change in x is the treatment-control difference, the slope is the difference between
the posttest means of the two groups.
Yi
Zi
Change in y
Why Why 11 Is the Mean Difference Is the Mean Difference
ininyi = 0 + 1Zi + ei
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
First, determine effect for each group:
yi = 0 + 1Zi + ei
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
yi = 0 + 1Zi + ei
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0 ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
yC = 0
ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
First, determine effect for each group:
For control group (Zi = 0):
For treatment group (Zi = 1):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
yC = 0
ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
For treatment group (Zi = 1):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
yC = 0
yT = 0 + 1(1) + 0
ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
For treatment group (Zi = 1):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
yC = 0
yT = 0 + 1(1) + 0
ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
First, determine effect for each group:
For control group (Zi = 0):
For treatment group (Zi = 1):
yi = 0 + 1Zi + ei
yC = 0 + 1(0) + 0
yC = 0
yT = 0 + 1(1) + 0
yT = 0 + 1
ei averages to 0across the group.
Why Why 11 Is the Mean Difference Is the Mean Difference
ininyi = 0 + 1Zi + ei
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yT = 0 + 1
yT
treatment
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC =
controltreatment
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC = (0 + 1)
controltreatment
Why Why 11 Is the Mean Difference Is the Mean Difference
inin
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC = (0 + 1) - 0
controltreatment
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC = (0 + 1) - 0
controltreatment
yT - yC = 0 + 1 - 0
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC = (0 + 1) - 0
controltreatment
yT - yC = 0 + 1 - 0
Why Why 11 Is the Mean Difference Is the Mean Difference
in in
Then, find the difference between the two groups:
yi = 0 + 1Zi + ei
yC = 0yT = 0 + 1
yT - yC = (0 + 1) - 0
controltreatment
yT - yC = 0 + 1 - 0
yT - yC = 1
ConclusionsConclusions
t-test, one-way ANOVA and regression t-test, one-way ANOVA and regression analysis all yield analysis all yield samesame results in this results in this case.case.
The regression analysis method utilizes The regression analysis method utilizes a a dummy variabledummy variable for treatment.for treatment.
Regression analysis is the most Regression analysis is the most generalgeneral model of the three.model of the three.