Transcript INFERENTIAL STATISTICS © LOUIS COHEN, LAWRENCE MANION & KEITH MORRISON STRUCTURE OF THE CHAPTER • Measures of difference between groups • The t-test (a.

INFERENTIAL STATISTICS
© LOUIS COHEN, LAWRENCE
MANION & KEITH MORRISON
STRUCTURE OF THE CHAPTER
• Measures of difference between groups
• The t-test (a difference test for parametric data)
• Analysis of variance (a difference test for
parametric data)
• The chi-square test (a difference test and a test of
goodness of fit for non-parametric data)
• Degrees of freedom (a statistic that is used in
calculating statistical significance in considering
difference tests)
• The Mann-Whitney and Wilcoxon tests (difference
tests for non-parametric data)
STRUCTURE OF THE CHAPTER
• The Kruskal-Wallis and Friedman tests (difference
tests for non-parametric data)
• Regression analysis (prediction tests for parametric
data)
• Simple linear regression (predicting the value of one
variable from the known value of another variable)
• Multiple regression (calculating the different
weightings of independent variables on a dependent
variable)
• Standardized scores (used in calculating
regressions and comparing sets of data with
different means and standard deviations)
REGRESSION
• Regression is a statistical technique of
modelling the relationship between variables.
• From knowing the values of one variable we can
predict the values of another variable
SIMPLE LINEAR REGRESSION
• Simple linear regression – the model includes
one explanatory variable (independent) and
one explained variable (dependent)
– The relationship between examinations
and stress
A SIMPLE REGRESSION
90
80
70
60
50
40
30
40
50
60
70
80
Score on final university examination
90
100
A SIMPLE REGRESSION (SPSS)
Model Summary
Model
1
R
.966a
R Square
.932
Adjusted
R Square
.932
Std. Error of
the Estimate
3.000
a. Predictors: (Constant), Hours per week on private study
ANOVAb
Model
1
Regression
Residual
Total
Sum of
Squares
12154.483
882.267
13036.750
df
1
98
99
Mean Square
12154.483
9.003
a. Predictors: (Constant), Hours per week on private study
b. Dependent Variable: Score on final university examination
F
1350.089
Sig.
.000a
A SIMPLE REGRESSION
Coefficientsa
Model
1
(Constant)
Hours per week
on private study
Unstandardized
Coefficients
B
Std. Error
22.201
1.432
.763
Standardized
Coefficients
Beta
.021
.966
t
15.504
Sig.
.000
36.744
.000
a. Dependent Variable: Score on final university examination
The beta weighting (ß) is ‘the amount of standard deviation unit
of change in the dependent variable for each standard deviation
unit of change in the independent variable’. Here the
standardized beta weighting is .966, i.e. it is highly statistically
significant (=0.000 in the ‘Sig.’ column); this means that for
every standard deviation unit change in the independent
variable (‘hours per week on private study’) there is .966 of a
unit rise in the dependent variable (‘score on final university
examination’) i.e. there is nearly a one-to-one correspondence.
MULTIPLE LINEAR REGRESSION
• The model is a linear equation with at least
two explanatory variables (independent) and
one explained variable (dependent)
10
USING MULTIPLE REGRESSION
• Multiple regression is useful in calculating the
relative weighting of two or more independent
variables on a dependent variable. Using the
beta () weighting, multiple regression calculates
how many standard deviation units are changed
in the dependent variable for each standard
deviation unit of change in each of the
independent variables.
• For example, let us say that we wished to
investigate the relative weighting of ‘hours per
week of private study’ and ‘motivation level’ as
independent variables acting on the dependent
variable ‘score on final university examination’.
Hours of
study per
week
Level of
motivation
Final
examination
score
USING MULTIPLE REGRESSION (SPSS)
Model Summary
Model
1
R
.969a
R Square
.939
Adjusted
R Square
.938
Std. Error of
the Estimate
2.852
a. Predictors: (Constant), Motivation level, Hours per
week on private study
The Adjusted R square is .938, i.e. the amount of
the dependent variable explained by the two
independent variables is very high (93.8%).
USING MULTIPLE REGRESSION
ANOVAb
Model
1
Regression
Residual
Total
Sum of
Squares
12247.676
789.074
13036.750
df
2
97
99
Mean Square
6123.838
8.135
F
752.796
Sig.
.000a
a. Predictors: (Constant), Motivation level, Hours per week on private study
b. Dependent Variable: Score on final university examination
The analysis of variance (ANOVA) is highly
statistically significant (=.000), i.e. the
relationship between the independent variable
and the dependent variable is very strong.
USING MULTIPLE REGRESSION (SPSS)
Coefficientsa
Model
1
(Constant)
Hours per week
on private study
Motivation level
Unstandardized
Coefficients
B
Std. Error
22.577
1.366
Standardized
Coefficients
Beta
t
16.531
Sig.
.000
.714
.024
.904
29.286
.000
.404
.119
.104
3.385
.001
a. Dependent Variable: Score on final university examination
USING MULTIPLE REGRESSION
The independent variable ‘hours per week of private study’ has the
strongest positive predictive power (=.904) on the dependent variable
‘score on final university examination’, and this is statistically significant
(the column ‘Sig.’ indicates that the level of significance, at .000, is
stronger than .001);
The independent variable ‘motivation level’ has strong positive predictive
power (=.104) on the dependent variable ‘score on final university
examination’, and this is statistically significant (the column ‘Sig.’ indicates
that the level of significance, at .001);
Though both independent variables have a statistically significant
weighting on the dependent variable, the beta weighting of the
independent variable ‘hours per week of private study’ (=.904) is much
higher than that of the independent variable ‘motivation level’ (=.104) on
the dependent variable ‘score on final university examination’, i.e. ‘hours
per week on private study’ is a stronger predictor of ‘score on final
university examination’ than ‘motivation level’.
USING MULTIPLE REGRESSION
The researcher can predict that, if the hours per week spent in private
study were known, and if the motivation level of the student was
known, then the likely score on the final university examination could
be predicted. The formula would be:
‘Score on final university examination’ =
( x ‘hours per week on private study’) + ( x ‘motivation level’)
In the example, the  for ‘hours per week on private study’ is 0.904,
and the  for ‘motivation level’ is 0.104. These are the relative
weightings of the two independent variables.
So, for example, for a student who spends 60 hours per week on
private study and has a high motivation level (9) the formula becomes:
‘Score on final university examination’ = (0.904 x 60) + (0.104 x 9)
= 54.24 + 0.936 = 55.176
For example, if the beta weighting were different, and for
different factors, the relationship between examination
mark, study time and intelligence could be:
Examination mark =  Study time +  Intelligence
Examination mark = 0.65 Study time + 0.30 Intelligence
A student with an intelligence score of 110 who studies for
30 hours per week will obtain the following examination
mark:
Examination mark
= (0.65 x 30) + (0.30 x 110)
= 19.5 + 33 = 52.5
If the same student studies for 40 hours then:
Examination mark
= (0.65 x 40) + (0.30) x 110
= 26 + 33 = 59
STRESS IN TEACHING (SPSS):
BETA WEIGHTINGS OF VARIABLES
Beta Coefficients
Standardized
Coefficients
Teacher voice and support
Workload
Benefits and rewards of teaching
Managing students
Challenge and debate
Family pressures
Considering leaving teaching
Emotions and coping
Burnout
Balancing work, family and
cultural expectations
Local culture
Stress from family
Stress reproducing stress
Control and relationships
Beta
.323
.080
.205
.116
.087
.076
.067
.044
.157
Significance
level
.000
.000
.000
.000
.000
.000
.000
.000
.000
.100
.000
.071
.058
.164
.092
.000
.000
.000
.000
STRESS IN TEACHING (SPSS)
(removing variables affects beta weightings)
Beta Coefficients
Standardized
Coefficients
Teacher voice and support
Workload
Benefits and rewards of teaching
Managing students
Challenge and debate
Considering leaving teaching
Emotions and coping
Burnout
Local culture
Stress reproducing stress
Control and relationships
Beta
.316
.096
.219
.114
.099
.102
.091
.156
.131
.162
.130
Significance
level
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
COLLINEARITY DIAGNOSTICS
(MULTI-COLLINEARITY)
• Multicollinearity: The correlation between each
independent variable should not be too high.
Collinearity diagnostics indicates the level of
correlation. If the collinearity is too high
between two variables then it may be advisable
to remove one.
• Multi-collinearity is tested by (a) Tolerance and
(b) the Variance Inflation Factor.
COLLINEARITY DIAGNOSTICS
(MULTI-COLLINEARITY)
Tolerance: ‘An indicator of how much of the variability
of the specified independent is not explained by the
other independent variables in the model . . . . If this
value is very small (less than .10), it indicates that the
multiple correlation with other variables is high,
suggesting the possibility of collinearity’ (Pallant,
2007: 156).
‘The VIF (Variance Inflation Factor), which is just the
inverse of the Tolerance Factor (1 divided by
Tolerance). VIF values above 10 would be a concern
here, indicating multicollinearity’ (Pallant, 2007: 156).
COLLINEARITY DIAGNOSTICS (SPSS)
(MULTI-COLLINEARITY)
Coefficientsa
Model
Unstandardized Standardized
Coefficients
Coefficients
Std.
Error
B
Beta
Collinearity
Statistics
t
Sig. Tolerance
VIF
1
(Constant)
22.577 1.366
16.531
.000
Hours per week on private
study
.714
.024
.904 29.286
.000
.655 1.528
Motivation level
.404
.119
.104 3.385
.001
.655 1.528
a. Dependent Variable: Score on final university examination
COLLINEARITY DIAGNOSTICS
(MULTI-COLLINEARITY)
In the example:
Hours per week on private study: Tolerance = .655
VIF = 1.528
Motivation level
Tolerance = .655
VIF = 1.528
There is no problem with multicollinearity.
SOME ASSUMPTIONS IN REGRESSION
• Random sampling;
• Ratio data;
• The removal of outliers: check outliers by
calculating the Mahalanobis distance (in SPSS) ;
• The supposed linearity of the measures is
justifiable;
• Interaction effects of independent variables (in
non-recursive models) are measured;
• The selection for the inclusion and exclusion of
variables is justifiable;
• The dependent variable and the residuals (the
distance of the cases from the line of best fit) is
approximately normally distributed;
SOME ASSUMPTIONS IN REGRESSION
• The variance of each variable is consistent
across the range of values for all other variables
(or at least the next assumption is true);
• The independent variables are approximately
normally distributed, the variation is even across
the levels/values of the variable
(homoscedasticity).
• Collinearity/multicollinearity is avoided.
• Regressions are only as robust as the variables
included, and the inclusion or removal of one or
more independent variables affects their relative
weightings on the dependent variable.
STEPWISE REGRESSION
• To find a model with predictive accuracy,
working with a limited number of independent
variables from a longer list of independent
variables, to determine which ones have a
statistically significant influence on the
dependent variables.
• Stepwise multiple regression enters variables
one at a time, in a sequence, to see which
adds to the explanatory power of a model, by
looking at its impact on the R-squared –
whether it increases the R-square value.
STEPWISE REGRESSION
Stepwise multiple regression enables the
researcher to see which variables have predictive
power and which do not, which to include and
which to exclude in an explanatory model.
LOGISTIC REGRESSION
• To enable the researcher to work with
categorical variables in a multiple regression
where the dependent variable is a categorical
variable.
• The independent variables may be
categorical, discrete or continuous.
PROCEDURE FOR MULTIPLE REGRESSION
1. Analyze  Regression  Linear.
2. Send over dependent variable to Dependent box.
3. Send over independent variables to Independent
box.
4. Click on Statistics. Tick the boxes Estimates,
Confidence Intervals, Model fit, Descriptives, Part
and partial correlations, Collinearity diagnostics,
Casewise diagnostics and Outliers outside 3
standard deviations. Click Continue.
5. Click on Options. Click on Exclude cases
pairwise. Click Continue.
6. Click on Plots. Send over *ZRESID to the Y box.
Send over *ZPRED to the X box. Click on Normal
probability plots. Click Continue.
7. Click on Save. Click the Mahalanobis box and the
Cook’s box. Click Continue.
8. Click OK.
EXAMINING MULTIPLE
REGRESSION SPSS OUTPUT
1. Check collinearity statistics:
a. Tolerance must be higher than .10;
b. VIF (Variance Inflation Factor) must not be
higher than 10.
2. Check normality, linearity and homoscedasticity:
a. Normality Probability Plot (Normal P-P Plot)
to have points going in a straight diagonal
line, bottom left to top right;
b. Scatterplot to be a rectangle with scores
concentrated in the centre (along the 0 point),
avoiding curvilinear or uneven distribution.
3. Check that the Cook’s Distance maximum
value is below 1 and that the Mahal. Distance
is lower than the critical value.
EXAMINING MULTIPLE
REGRESSION SPSS OUTPUT
4. Check the Adjusted R Square.
5. Check ANOVA and its significance level.
6. Check the Standardized Beta Coefficients
and their significance levels.
7. Square each Parts correlation coefficient to
see the contribution of each variable to the
total Adjusted R Square (i.e. how much of the
total variance in the dependent variable is
explained by each independent variable).
COLLINEARITY DIAGNOSTICS
(MULTI-COLLINEARITY)
1.
2.
3.
4.
5.
6.
7.
Analyze 
Regression 
Linear 
Statistics 
Click ‘Collinearity diagnostics’
Click ‘Continue’
Continue with the multiple regression.
PROCEDURES FOR STEPWISE
MULTIPLE REGRESSION IN SPSS
• ‘Analyze’  ‘Regression ‘ ‘Linear ‘ Enter
dependent and independent variables  In the
‘Method’ box, change ‘Enter’ to ‘Stepwise’ 
Click ‘OK’.
PROCEDURES FOR LOGISTIC
REGRESSION IN SPSS
Analyze  Regression  Binary Logistic  Insert
dependent variable in the ‘Dependent’ box  Insert
independent variables into the ‘Covariates’ box 
Click on ‘Categorical’  Move first categorical
variable into the ‘Categorical Covariates’ box 
Click the radio button ‘First’  Click the ‘Change’
button  Repeat this for every categorical variable
 Click ‘Continue’ to return to the first screen 
Click ‘Options’  Click the boxes ‘Classification
plots’, Hosmer-Lemeshow goodness of fit’,
‘Casewise listing of residuals’ and ‘CI for Exp(B) 
Click ‘Continue’ to return to first screen  Click ‘OK’.
THE NEED FOR A STANDARDIZED
SCORE
• A child tells his parents that he scored a mark of
75 for a maths test; his parents scold him.
• A child tells his parents that he scored a mark of 2
for a history test; his parents praise him.
• A child tells his parents that he scored a mark of
25 for an English test and a mark of 60 for a
Physics test; his parents praise him for both.
• A child tells his parents that he scored a mark of
80 for a Geography test and a mark of 120 for a
Chemistry test; his parents scold him for both.
THE NEED FOR A STANDARDIZED
SCORE
• We need to know how to judge whether a
mark is high or low.
• We need to be able to compare marks
between one test and another.
• Therefore we need to know the scale of the
marks, the range of the marks, the mean of
the marks, and the distribution of the marks
either side of the mean.
THE NEED FOR A STANDARDIZED
SCORE
• We need to know how to compare marks
from a test which:
– uses one scale with marks from a test
which uses another scale;
– has one range of marks with marks from a
test that has another range of marks;
– has a mean which is different from the
mean of another test;
– Has a distribution around the mean which
is different from the distribution of another
test.
THE Z-SCORE
(STANDARDIZED SCORE)
• Standardizing scores lets us judge whether a
mark is high or low.
• Standardizing scores lets us compare marks
between one test and another when two
different tests have different scales, range,
means and distributions around the mean.
Z-SCORES
• z-scores have the same mean and standard
deviation, even though the original sets of
scores had different means and standard
deviations, i.e. z-scores let you compare
fairly.
• A z-score tells us how many standard
deviations someone’s scores are above or
below the mean.
Z-SCORES
• To calculate the z-score subtract the mean from
the raw score and divide that answer by the
standard deviation.
the actual score  the m eanof the sam ple xi  x
z

standarddeviationof thesample
s
• For example if the raw score is 15, the mean is
10, and the standard deviation is 4, then 15-10
= 5 and 5  4 = 1.25.
• Here z-score tells us that the person’s score is
1.25 standard deviations above the mean.
Is that score good or bad? How good or
bad is it?
Two standard
deviations either side
of the mean accounts
for 95.4% of the
population.
The mean
One standard
deviation either
side of the mean
accounts for
68.3% of the
population.
STANDARDIZED SCORE (Z-SCORE)
• A z-score of +1.4 indicates that someone is 1.4
standard deviations above the mean.
• A z-score of -1.4 indicates that someone is 1.4
standard deviations below the mean.
• If the z-score is positive, it indicates that the value is
above the mean.
• If the z-score is negative, it means that the value is
below the mean.
Is that z-score good or bad? How good or bad is it?
We need to know about the probability of a certain
value falling into a certain range of value.
The normal
curve lets us
interpret the
probability of a
score falling
into a certain
range of
scores/values.
68% of the population lie between -1 and +1 standard deviations
95% of the population lie between -2 and +2 standard deviations
99% of the population lie between -1 and +1 standard deviations
The normal curve lets us interpret the probability of a
score falling into a certain range of scores/values.
Let us say that, for 1200 people:
The mean = 35
The standard deviation = 13
If a member of the group says he is 61 years old
then it is clear that this person is much older
than the average. But how much older? To be
exact, we can convert his score into a z-score.
61  35
z
2
13
61  35
z
2
13
• This tells us that 61 is 2 standard deviations
above the mean.
• Refer to the ‘areas under the standard
normal curve’ table (in statistics textbooks),
for z = 2, the ‘area under curve beyond one
point’ is 0.023.
• The proportion of people that are 61 years of
age or more is only 2.3% of the total.
For example if z-score is 1.25.
Is that score good or bad? How good or bad
is it?
• Refer to the ‘areas under the standard
normal curve’ table (in statistics textbooks),
for z = ±1.25, the ‘area under curve beyond
one point’ is 0.1056.
• The proportion of people who score 1.25 or
more is only 10.56% of the total.
So, the score is very good.
An online calculator of area under the curve for
standardized scores is at:
www.danielsoper.com/statcalc/calc02.aspx
This calculator gives the cumulative area under the
curve (a figure as a decimal fraction that is less
than 1 (let us call it X). To find the area under the
curve beyond that one point simply subtract this
figure from 1 (the formula, then is 1-X).
CALCULATING Z-SCORES WITH SPSS
• Click on ‘Analyze’  ‘Descriptive Statistics’ 
Descriptives’.
• Send over the variables to ‘Variables’  ‘Click the
box ‘Save standardized values as variables’  Click
‘OK’ Two new variables will be created.
T-SCORES
• Some people are uncomfortable with z-scores, as
they don’t like negative scores and they do not like
an average being 0.
• To overcome this, z-scores can be converted to Tscores. To convert a z-score to a T-score, multiply
the z-score by 10 and add 50 to the result.
• For example a z-score of .5, multiplied by 10 gives
5, and then, with 50 added, gives 55. The T-score
is 55.
• Many IQ tests and standardized tests convert zscores. For example a common conversion in IQ
tests is to multiply the z-score by 15 and add 100.
So a z-score on an IQ test might be .5, multiplied
by 15 gives 7.5, with 100 added gives 107.5, i.e.
the IQ z-score converts to a T-score of 107.5.
THE MCCALL T-SCORE
• The McCall T-score has a mean of 50 and a
standard deviation of 10:
– McCall T-score = 50±(z-score x 10)
• For the ± sign, the part in brackets should be
added to the 50 if the z-score is positive (i.e. if
the raw score is above the mean) and
subtracted if the z-score is negative (i.e. if the
raw score is below the mean).