METHODS OF KNOWINGCharles Pierce
1.TENACITY
2. AUTHORITY
3.INTUITION
4. SCIENCE
TENACITY
• We’ve always done it this way
• “When I was a kid, school was…”
AUTHORITY
• X says…
• The Bible (Koran, Confucius, Vedic Script)
says…
• The Founding Fathers meant…
• Dr. Saxon’s method says…
INTUITION
•
•
•
•
•
It is only common sense that…
It is obvious…
Anyone can see…
Everyone agrees…
The majority says…
SCIENCE
• DISCIPLINED INQUIRY
1. Self-correction
2. Objective-orientationintersubjective
confirmability
3. Active and passive
exploration
Description-Explanation
•
•
•
•
•
•
•
• Specificity
(uniqueness
phenomenology
Naturalistic
observe existing
Universality
generalizability)
logical positivism
Scientific
experimentation,
observation
multiple realities objective reality
(social,personal)
(multiple
perceptions)
Constructivism
Cognitive
science
ideological linkage apolitical orientation
(Social justice etc.)
EXPERIMENTATION
• Systematic intervention into natural
system
• Careful observation of conditions:
- Before/after
- Comparison with different case/system
• Attribution of CAUSE to the intervention
CAUSE
• Does not exist: Hume, Bertrand Russell
(extreme logical positivist: relationships
among natural variables can be described
by mathematical statements)
• Essentialists: cause exists if and only if
presence of the cause results in an
EFFECT; the effect occurs only when the
cause is present
CAUSE
• Empiricists: J. S. Mill argued 3
necessary conditions:
AGREEMENT: the effect occurs after the effect
DIFFERENCE: the effect does not occur when the
effect is absent in general
CONCOMITANT VARIATION: support for causal
relationship is greater when AGREEMENT and
DIFFERENCE are close in time
CAUSE- Nonmanipulable &
Manipulable
• Some causes can be observed but not
manipulated due to ethical, natural, or
practical considerations
• eg. Gender causes differences in career
interest
• eg. PTSS causes hallucinations
• eg. Divorce causes insecurity in children
CAUSE- Molar and Molecular
• Molecular causes- focus on small scale
units, subparts
• Molar causes- focus on larger systems
that are made up of molecular causes
eg: Teaching a student to decode words
causes improvement in simple
comprehension - molar
Teaching a student to sound out
beginning letters causes decoding gains
EXPERIMENTS
• RANDOMIZED
– R. A. Fisher, 1923: split plot randomization of
treatments; treatment as cause vs. chance
• NONRANDOMIZED
– Internal validity threats as alternative causal
explanations
NATURAL EXPERIMENTS
• Time series experiments
– Occurence of new condition in one
place/group
– Nonoccurence in a similar place/group
NATURAL EXPERIMENTS
700
600
500
400
300
200
100
1
13
7
25
19
Case Number
37
31
49
43
61
55
73
67
85
79
97
91
103
NATURAL EXPERIMENTS
700
600
500
400
300
200
100
1.00
13.00
7.00
MONTH
25.00
19.00
37.00
31.00
49.00
43.00
61.00
55.00
73.00
67.00
85.00
79.00
97.00
91.00
103.00
CONSTRUCT VALIDITY
• Single study as data point
• Variables in study represent constructs,
rarely define them
• Study results often interactive with
construct definition of the study- revision
based on the conduct, findings of the
study
EXTERNAL VALIDITY
• 3 Constructs:
– Population: persons/groups
– Ecology: settings
– Time: absolute or relative time
SAMPLING ISSUES
• Formal random sampling theory
– Statistical theory of estimation
• Purposive sampling
– ignore sampling theory, focus on particulars
• Convenience sampling
– availability of participants/subjects
Grounded Theory of
Generalization
• Surface similarity of sample to population
• Rule out irrelevant variables, conditions
• Identify limits to generalizationdiscriminations
• Interpolate and extrapolate data
• Develop causal explanations: covariances
and directional relationships
CRITIQUES AND CRITICS
• Kuhn- theories are incommensurable
– theory vs. observation/data
•
•
•
•
Collins- science is social construction
Trust vs. skepticism in science
External reality of data vs. theories
Science as a preferred human
predisposition:
– evolutionary result?
Multiple regression analysis
•
•
•
•
•
Two or more interval scale predictors
Single interval dependent variable
Predictors “known without error”
Model: y = b1x1 + b2x2 + b0 + e
Create predicted score



yhat = b1x1 + b2x2 + b0 where  means
estimate of the model
• OLS estimation is most commonly used
Multiple regression analysis
(MRA)
• The test of the overall hypothesis that y is
unrelated to all predictors, equivalent to
• H0: 2y123… = 0
• H1: 2y123… = 0
• is tested by
• F = [ R2y123… / p] / [ ( 1 - R2y123…) / (n – p – 1) ]
• F = [ SSreg / p ] / [ SSe / (n – p – 1)]
ANOVA table for MRA
SOURCE
x1, x2…
e (residual)
total
df
Sum of Squares
Mean Square F
p
SSreg
SSreg / p
SSreg/ 1
SSe /(n-p-1)
n-p-1 SSe
SSe / (n-p-1)
n-1
SSy / (n-1)
SSy
• Table 8.2: Multiple regression table for Sums of Squares
Multiple regression analysis
predicting Depression
Model Summary
Model
1
R
.774a
R Sq uare
.600
Adjusted
R Sq uare
.596
Std. Error of
the Estimate
6.120
a. Predictors: (Constant), t11, t9, t10
ANOVAb
Model
1
Reg ression
Residual
Total
Sum of
Squares
21819.235
14571.498
36390.733
df
3
389
392
Mean Square
7273.078
37.459
a. Predictors: (Constant), t11, t9, t10
b. Dependent Variable: t6
LOCUS OF CONTROL, SELF-ESTEEM, SELF-RELIANCE
F
194.162
Sig .
.000a
VENN DIAGRAMS
• Venn diagrams are “heuristic” only for
more than two predictors
• They do not correctly separate sums of
squares for 3 or more predictors (cannot
represent 4 or more dimensions correctly
in flat 2-D space
• Still give us an idea of the rationale for
predictor additions to prediction
SSreg
ssx1
SSy
SSe
ssx2
Fig. 8.4: Venn diagram for multiple
regression with two predictors and
one outcome measure
Type I and III Sums of Squares
• Type I sums of squares are the SS
accounted for by a predictor in a specific
order: the analyst specifies the order or
allows the MRA program to pick the order
– Forward regression: best predictor (most SS
accounted for) is included first; second
predictor is the one that accounts for the most
additional SS
• Type III sums of squares are the unique
SS accounted for by a predictor
Type I
ssx1
SSx1
SSy
SSe
SSx2
Type III
ssx2
Fig. 8.5: Type I contributions
Type III
ssx1
SSx1
SSy
SSe
SSx2
Type III
ssx2
Fig. 8.6: Type IIII unique
contributions
Multiple Regression ANOVA table
SOURCE
•
•
•
•
•
•
•
Model
•
df
2
Sum of Squares Mean Square
(Type I)
SSreg
SSreg / 2
x1
1
SSx1
SSx1 / 1
x2
1
SSx2  x1
SSx2  x1
e
n-3
SSe
SSe / (n-3)
total
n-1
SSy
SSy / (n-3)
F
SSreg / 2
SSe / (n-3)
SSx1/ 1
SSe /(n-3)
SSx2  x1/ 1
SSe /(n-3)
Table 8.3: Multiple regression table for Sums of Squares of each predictor
Path Models for MRA
• Predictors are called exogenous variables
– Exogenous variables never have a straight path arrow
directed toward them
– They may have curved “correlation” arrows
connecting them to other exogenous var’s
• Dependent variables are called endogenous
variables
– They always have at least one straight path arrow
directed toward them
– Endogenous variables may be predictors of other
endogenous variables in more complex models
Accounting for correlation in path
models
• The correlation between any two variables
is the sum of the path effects between
them
• Any path effect is a unique pathway that
may pass though any number of other
exogenous and/or endogenous variables
• The total path effect is computed by
multiplying all path coefficients together
along a path
PATH DIAGRAM FOR REGRESSION
X1
 = .5
.387
r = .4
Y
X2
e
 = .6
r(x2,y) = .6 (direct path)
r(x1,y) = .5 (direct path)
+ .4 x .6 (unanalyzed path
= .74
+ .4 x .5 (unanalyzed path
= .80
PATH DIAGRAM FOR REGRESSION
X1
 = .5
.387
r = .4
Y
X2
 = .6
e
R2 = .742 + .82 - 2(.74)(.8)(.4)
R2 =[ rx1y + rx2y
- 2(rx1x2)(rx1y + rx2y) ]
(1-. rx1x22)
(1-.42)
= .85
Estimating regression weights
• Normal equations are computed (OLS)
• Least squares estimates are BLUES
• “Raw” or unstandardized regression
weights are used to predict an outcome
score for a specific set of predictor values
• Statistical test for signficance for a bweight is t-distributed with n-p-1 degrees
of freedom (n=number of cases, p=#
predictors)
Estimating regression weights
• SPSS Regression in Analyze provides b
estimates,also “standardized” estimates
• Standardized estimates are also called
beta weights
– Beta weights are regression coefficients when
all predictors and the dependent variable
have mean zero and variance 1 (z-scores)
• Only raw weight t-statistics can be
interpreted in hypothesis tests formally
Depression
Coefficientsa
Model
1
(Constant)
t9
t10
t11
Unstandardized
Coefficients
B
Std. Error
51.939
3.305
.440
.034
-.302
.036
-.181
.035
Standardized
Coefficients
Beta
.471
-.317
-.186
t
15.715
12.842
-8.462
-5.186
Sig .
.000
.000
.000
.000
a. Dependent Variable: t6
e
LOC. CON.
.4 = .63
.471
-.317
DEPRESSION
SELF-EST
SELF-REL
-.186
R2 = .60
Shrinkage R2
• Different definitions: ask which is being
used:
– What is population value for a sample R2?
• R2s = 1 – (1- R2)(n-1)/(n-k-1)
– What is the cross-validation from sample to
sample?
• R2sc = 1 – (1- R2)(n+k)/(n-k)
Estimation Methods
• Types of Estimation:
– Ordinary Least Squares (OLS)
• Minimize sum of squared errors around the
prediction line
– Generalized Least Squares
• A regression technique that is used when the
error terms from an ordinary least squares
regression display non-random patterns such
as autocorrelation or heteroskedasticity.
– Maximum Likelihood
Maximum Likelihood Estimation
• Maximum likelihood estimation
• There is nothing visual about the maximum likelihood
method - but it is a powerful method and, at least for
large samples, very precise
• Maximum likelihood estimation begins with writing a
mathematical expression known as the Likelihood
Function of the sample data. Loosely speaking, the
likelihood of a set of data is the probability of
obtaining that particular set of data, given the chosen
probability distribution model. This expression
contains the unknown model parameters.
• The values of these parameters that maximize the
sample likelihood are known as the Maximum
Likelihood Estimatesor MLE's. Maximum likelihood
estimation is a totally analytic maximization
procedure.
•
Maximum Likelihood Estimation
L
Maximum L at b= 1.345
(likelihood
function)
0
b values
1
2
3
Maximum Likelihood Estimation
MLE's
and Likelihood Functions generally have very
•
desirable large sample properties:
they become unbiased minimum variance estimators as the sample
size increases
they have approximate normal distributions and approximate sample
variances that can be calculated and used to generate
confidence bounds
likelihood functions can be used to test hypotheses about models and
parameters
With small samples, MLE's may not be very precise and may even
generate a line that lies above or below the data points
There are only two drawbacks to MLE's, but they are important ones:
1 With small numbers of failures (less than 5, and sometimes less than
10 is small), MLE's can be heavily biased and the large
sample optimality properties do not apply
2 Calculating MLE's often requires specialized software for solving
complex non-linear equations. This is less of a problem as
time goes by, as more statistical packages are upgrading to
contain MLE analysis capability every year.
Outliers
• Leverage (for a single predictor):
• Li = 1/n + (Xi –Mx)2/ x2 (min=1/n, max=1)
• Values larger than 1/n by large amount
should be of concern
 – Yi)
 2 / [(k+1)MSres]
• Cook’s Di = (Y
– the difference between predicted Y with and
without Xi
Outliers
• In SPSS Regression, under the SAVE
option, both leverage and Cook’s D will be
computed and saved as new variables
with values for each case
t12
t13
t14
63
42
41
56
56
41
77
52
39
30
53
52
55
42
59
48
50
50
55
55
60
39
39
65
39
45
60
55
39
80
80
65
44
68
46
52
57
41
65
54
65
60
68
68
41
68
68
46
COO_1 LEV_1
.03855
.02422
.02065
.01915
.01696
.01689
.01525
.01448
.01425
.01242
.01133
.01060
.01047
.00918
.00907
.00885
.01520
.04943
.02010
.02349
.01056
.02435
.01520
.01607
.02289
.01346
.03147
.00693
.00512
.02459
.01098
.00160
Might
reanalyze
with these
data
points
omitted