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PSYC512: Research Methods
Lecture 19
Brian P. Dyre
University of Idaho
PSYC512: Research Methods
Lecture 19 Outline

Inferential Statistics
 Testing for differences vs. relationships
 Analyzing frequencies
 Analyzing differences between means
PSYC512: Research Methods
Using Inferential Statistics
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Which Statistic?
 The statistical decision tree Howell Figure 1.1
Testing for relationships vs. differences (a false distinction)
 Relationships: assessing the strength of relationship
between measured (dependent) variables
 Differences: comparing different groups or treatments
on some measurement
 But what causes those differences? The relationship
between the independent variable defining the
groups or treatment and the dependent variable
 Hence, testing for differences is really testing the
relationship between the IV and DV
PSYC512: Research Methods
Analyzing Differences Between
Treatments
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Nominal and Ordinal Frequency Data
 “Success vs. Failure” - Binomial Distribution and The Sign Test
 Multiple categories (> 2) Multinomial distribution and Chi-square
 Multidimensional categories: Chi-square contingency tables
Integral and Ratio Data
 2 treatments or groups – t-test
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Comparing two independent samples  HW3
Comparing two correlated (or paired samples)  HW4
More than 2 treatments or groups – ANOVA
More than 2 independent variables – multifactor ANOVA– HW5
2 or more dependent variables (or repeated measures) –
MANOVA
Covariate  ANCOVA – HW5
Relations between measures

Correlation or Regression
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 5)
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Bernoulli Trials: series of independent trials that result in one of two
mutually exclusive outcomes
 E.g. coin flips, gender of babies born, increase of decrease in a
measure after application of a treatment
The Binomial Distribution
p ( X )  C XN p X q ( N  X ) where,
C XN  The number of combinations of N things taken X at a time 
N!
p X q ( N  X ) where
X !( N  X )!
p ( X )  The probabilit y of X successes
N  The number of trials
p  The probabilit y of " success" on any one trial
q  (1  p )  The probabilit y of " failure" on any one trial
p( X ) 
PSYC512: Research Methods
N!
, hence
X !( N  X )!
Analyzing Frequencies
(Howell, Chapter 5)
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N!
Using the binomial distribution
p( X ) 
p X q(N  X )
X !( N  X )!
 Mean number of successes = Np
 Variance in number of successes = Npq
Testing Hypotheses using the binomial distribution: The Sign Test
 Ho is typically p= q = .50 (50-50 chance of success of failure), but
that doesn’t have to be the case
 H1 is typically p ≠q
 Plug in values for N, X, p, and q and p(X) directly provides the
probability that the pattern of data could result given the null
hypothesis is true
 Sum the probabilities p(X) for all number >= X to get the total
probability of finding p(>=X)
 Important: The sign test takes into account direction of differences
but not magnitude
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 5)

What about multiple (more than 2) possible outcomes?
 Multinomial distribution
N!
p( X 1 , X 2 ,... X k ) 
p X 1 p X 2 ... p X k where,
X 1! X 2 !... X k !
where
p( X 1 , X 2 ,... X k )  The probabilit y of frequency X in each category, k
N  The number of trials
p X k  The probabilit y of observation X being in category k on any one trial
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 5)
p( X 1 , X 2 ,... X k ) 
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Using the multinomial distribution
 Mean Xk = NpXk
 Variance in Xk = NpXk (1-pXk)
N!
p X1 p X 2 ... p X k
X 1! X 2!...X k !
Testing Hypotheses using the multinomial distribution:
 Ho is typically pX1= pX2 … = pXk = 1/k (each outcome has the
same chance), but that doesn’t have to be the case
 H1 is typically pX1 ≠ pX2 …≠ pXk
 Plug in values for N, X, and pX, and p(X1, X2…Xk) directly provides
the probability that this particular pattern of data could result
given the null hypothesis is true
 Must sum the probabilities for all patterns that deviate equal to or
more to get the total probability – time consuming!
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 6)
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Easier Alternative to
Multinomial distribution:
Chi-square (c2) test
Compare computed value
of c2 to value of c2
distribution with df=k-1
Expected frequencies for
the null hypothesis
typically = N/k, where N
is the total number of
observations
c
(Oi  Ei )

Ei
i 1
k
2
k 1
k is the number of
2
categories in the variable
O is the observed frequency
for each category
E is the expected frequency
for each category
i is the category index
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 6)
R
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c2 with
Using
multiple
dimensions: contingency
tables—frequencies of one
dimension are contingent on
the other dimension
Eij = RiCj/N
N is the total number of
observations
Compare computed value of c2
to value of c2 distribution with
df=(R-1)(C-1)
C
c (2R 1)(C 1)  
i 1 i 1
(Oij  Eij ) 2
Eij
R is the number of categories in
the dimension defined by the
rows of the table
C is the number of categories in
the dimension defined by the
columns of the table
O is the observed frequency for
each category
E is the expected frequency for
each category
i and j are category indices
PSYC512: Research Methods
Analyzing Frequencies
(Howell, Chapter 6)

Assumptions of the c2 test
 Each observation is independent
 Inclusion of non-occurrences
PSYC512: Research Methods
z-tests, t-tests
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s of population is known: z
s of population is estimated as s: t
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df = N-1
zX 
X 
sX

X 
s/ N
X  X 
t X ( N  1) 

sX
s/ N
D 0
D
Comparing 2 paired (or correlated) samples
t X ( N  1) 

 Difference scores
sD
sD / N
 Df = N -1
Comparing 2 independent samples
 df = n1 + n2 – 2
 Unequal sample sizes, heterogeneity of
variance, and pooled variances
PSYC512: Research Methods
t X (n1  n2  2) 
( X1  X 2 )
s12 s22

n1 n2
ANOVA (F Statistic)
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Used when comparing more than 2 means or 2 or more factors
Assumptions
 Homogeneity of variance
 Normality
 Independence of observations
MS treatment
Between Groups comparisons
F (k  1, k (n  1)) 
MS error
 k = number of means compared
 n = number of Ss in group
Repeated Measures
MS treatment
F
(
k

1
,
k
(
n

1
))

 Error term is interaction of error with
MS s x error
subject random variable
PSYC512: Research Methods
Interpreting SPSS output
PSYC512: Research Methods