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STAT 460
Lecture 15: Review
10/27/2004
1) Goal: Draw valid conclusions (with a degree of uncertainty) about the relationship
between explanatory variable(s) and an outcome variable in the face of the limited
experimental resources and the randomness characteristic of the world.
2) One detailed example: Experiment to study the effects of a “teamwork”vs. a
“technical” intervention on productivity improvement in small manufacturing firms.
3) Principles of Statistics
a) A statistical view of the world
The Population
A Sample
Contains n randomly chosen subjects.
Observed expl. & outcome variables.
Some statistics for this sample:
Contains N subjects.
Unobserved variables.
Some of this population’s parameters:

μcontrol β0control βempcnt,control

μteamwork β0team βempcnt,team

μtechnical β0tech βempcnt,tech
σ2rx only σ2rx,empcnt σ2rx only, multiple teachers
X control
̂ control ̂ empcnt,control
X teamwork
̂ 0team ̂ empcnt,team
X technical
̂ 0tech
s2rx only
s2rx,empcnt
̂ empcnt,tech
s2rx only, mult
b) Hypothesis testing
i) ANOVA: Let μi for i in 1 to k be the population mean for the outcome for
group i. H0: μ1=…=μk vs. H1: at least one pop. mean differs
ii) Regression: Let β0 be the population value of the intercept and β1 be the
population value of the slope of a line representing the mean outcome for a
range of values of explanatory variable 1. H0: β0=0 vs. H1: β0≠0 H0: β1=0
vs. H1: β1≠0 etc.
iii) ANOVA decision rule: reject H0 if Fexperimental>Fcritical equiv. to p<α
iv) Regression decision rules: reject H0 if |t|>tcritical equiv. to p<α
v) If we choose α=0.05, then when the null hypothesis is true we will falsely
reject the null hypothesis (make a type 1 error) only 5% of the time. (This
only guarantees that we will correctly reject the null at least 5% of the time
when the null is false.)
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c) Characteristics of problems we can deal with so far:
i) Quantitative outcome variables and a categorical explanatory variable
(ANOVA)
ii) Quantitative outcome variable (regression) or one of each (ANCOVA form of
regression)
iii) We require the following assumptions to make a specific model that we can
analyze:
(1) the outcome is normally distributed with the same variance at each set of
explanatory variable values;
(2) the subjects (actually, errors) are independent;
(3) the explanatory variables can be measured with reasonable accuracy and
precision.
(4) For regression, the relationship between quantitative explanatory variables
and the outcome is linear on some scale.
(5) For one-way ANOVA, the k population means can have any values, i.e.
there is no set pattern of relationship between the outcome and the
explanatory variables.
d) Examples of data we have not learned to deal with:
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e) Experiments vs. observational studies
i) In experiments, the treatment assignments are controlled by the experimenter.
Randomization balances confounding in experiments.
ii) In randomized experiments, association can be interpreted as causation.
iii) In observations studies, causation can be in either direction or due to a third
variable.
f) The assumptions (fixed x, independent errors, normality with the same variance
(σ2) for any set of explanatory variables, plus linearity for regression models) are
needed to calculate a “null sampling distribution” for any statistic.
i) This tells the frequency distribution of that statistic over repeated samples
when the null hypothesis is true, thus allowing calculation of the p-value.
ii) The alternate sampling distributions require more information and more
difficult calculations.
iii) Note that we do not require normality or equal variance for explanatory
variables.
g) The p-value is the probability of getting a sampling statistic (e.g. ANOVA F or
regression t) that is as extreme or more extreme that the one we got for this
experiment if the null distribution were true.
h) The one-way ANOVA F value has a particularly nice interpretation. MSbetween in
the numerator estimates σ2 from the model if the null hypothesis is true, and
something bigger otherwise. MSwithin in the denominator estimates the σ2
regardless of whether the null hypothesis is true or not. So F is around 1 when the
null hypothesis is true and bigger otherwise.
i) Degrees of freedom count the amount of information in a statistic by subtracting
the number of “constraints” from the number of component numbers in the
statistic. Use df to check that an analysis was set up correctly or to obtain certain
information about an analysis from statistical output. E.g. the df in MSbetween is k1 (one less than the number of groups being compared).
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4) Experimental Design concepts
a) Generalizability
i) Assure that the population from which samples of subjects could be drawn is
not too restricted. Assure that treatment and environmental conditions are not
too restrictive.
ii) Be more and more careful in stating conclusion as they apply to populations
less and less restrictive than where the sample came from.
iii) Balance against increased control for increased power.
b) Power is the chance that the null hypothesis will be rejected for some specific
alternate hypothesis. In one-way ANOVA, e.g., power is increased by increasing
MS between  2  n  effects
the F statistic, F 
.

MS within
2
i) Don’t study small effects
ii) Decrease σ2 by decreasing subject, treatment, environmental and measurement
variation
iii) Increase sample size
c) Interpretability: don’t make alternate explanations easy to defend
i) Use a control
ii) Use blinding of subject and/or experimenter
iii) Use randomization to prevent confounding
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5) EDA is used to find mistakes, become familiar with data and coding, anticipate
results, and roughly check for appropriate models.
a) All models: check descriptive statistics for quantitative variables and frequency
tables for categorical variables.
b) ANOVA: side-by-side boxplots of outcome by the categorical explanatory
variable.
2.0
1.5
1.0
Productivity
.5
0.0
-.5
N=
30
35
control
teamwork
25
technical
Treatment
c) Regression: scatterplot with outcome on the y-axis.
d) ANCOVA: scatterplot with separate symbols for each group.
2.0
1.5
1.0
.5
Productivity
Treatment
0.0
technical
teamwork
-.5
control
0
20
40
60
80
Employee Count
6
100
120
140
6) Analysis details
a) ANOVA
Response: Productivity
Treatment: Control (0), Teamwork (1), Technical (2)
The MEANS Procedure
Analysis Variable : Productivity
N
Treatment
Obs
Mean
Std Dev
N
Minimum
Maximum
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0
30
0.5533333
0.2823709
30
-0.0500000
0.9900000
1
35
0.6237143
0.3365402
35
-0.1100000
1.2200000
2
25
1.0324000
0.2769067
25
0.4400000
1.4800000
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The ANOVA Procedure
Class Level Information
Class
Levels
Treatment
Values
3
0 1 2
Number of observations
90
The ANOVA Procedure
Dependent Variable: Productivity
Source
DF
Model
2
Error
Corrected Total
Sum of
Squares
Mean Square
F Value
Pr > F
3.59417575
1.79708787
19.54
<.0001
87
8.00333981
0.09199241
89
11.59751556
R-Square
Coeff Var
Root MSE
prtivity Mean
0.309909
42.49257
0.303303
0.713778
Source
DF
Treatment
2
Anova SS
3.59417575
ANOVA
Productivity
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Mean Square
F Value
1.79708787
19.54
Pr > F
<.0001
b) Regression and ANCOVA
The REG Procedure
Model: MODEL1
Dependent Variable: Productivity
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
5
84
89
4.66103
6.93649
11.59752
0.93221
0.08258
Root MSE
Dependent Mean
Coeff Var
0.28736
0.71378
40.25939
R-Square
Adj R-Sq
F Value
Pr > F
11.29
<.0001
0.4019
0.3663
Parameter Estimates
Variable
Label
Intercept
EmployeeCount
Teamwork
TechSkill
TeamEmployee
TechEmployee
Intercept
Teamwork
TechSkill
DF
1
1
1
1
1
1
Parameter
Estimate
Standard
Error
0.79282
-0.00578
0.01054
0.17951
0.00300
0.00686
0.09866
0.00202
0.14145
0.13708
0.00244
0.00243
t Value
8.04
-2.87
0.07
1.31
1.23
2.82
Standardized
Pr > |t| Estimate
<.0001
0.0052
0.9408
0.1939
0.2220
0.0060
Parameter Estimates
Variable
Label
Intercept
EmployeeCount
Teamwork
TechSkill
TeamEmployee
TechEmployee
Intercept
Teamwork
TechSkill
DF
1
1
1
1
1
1
9
95% Confidence Limits
0.59662
-0.00978
-0.27075
-0.09309
-0.00185
0.00202
0.98902
-0.00177
0.29183
0.45211
0.00785
0.01169
0
0.58066
0.01431
0.22398
0.32180
0.63719
7) Residual analysis
.8
.6
.4
Unstandardized Residual
.2
0.0
Treatment
-.2
technical
-.4
teamwork
-.6
control
-.8
Total Population
0
20
40
60
80
100
120
140
Employee Count
.8
.6
.4
Unstandardized Residual
.2
0.0
Treatment
-.2
technical
-.4
teamwork
-.6
control
-.8
Total Population
.8
1.0
1.2
1.4
1.6
Log10(Employee Count)
10
1.8
2.0
2.2
Normal P-P Plot of Regression Standardized Residual
Dependent Variable: Productivity
1.00
Expected Cum Prob
.75
.50
.25
0.00
0.00
.25
.50
Obs erved Cum Prob
11
.75
1.00
8) Transformation and Interpretation
Coefficients(a)
Unstandardized
Coefficients
Model
1
B
Standardized
Coefficients
Std. Error
Beta
t
Sig.
(Constant)
1.417
.284
4.989
.000
Teamwork
-.035
.409
-.047
-.085
.932
Technical Skills
-.618
.388
-.772
-1.593
.115
.126
.248
.303
.509
.612
.710
.243
1.471
2.919
.005
-.564
.182
-.495
-3.091
.003
F
12.529
Sig.
.000
Teamwork/Log1
0(Employee
Count)
interaction
tech/Log10(Empl
oyee Count)
interaction
Log10(Employee
Count)
a Dependent Variable: Productivity
Model Summary
Model
1
R
.654
R Square
.427
Adjusted
R Square
.393
Std. Error of
the Estimate
.28122
ANOVA
Model
1
Regres sion
Residual
Total
Sum of
Squares
4.954
6.643
11.598
df
Mean Square
.991
.079
5
84
89
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