Design & Statistics III
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Univariate – one variable
Bivariate – two variables
Skewness and kurtosis should be between +1 and -1
Skewness is significant if its value is larger than standard error x 2
Monotonic – does y rise or fall consistently as x rises? A u-shaped
relationship is not monotonic.
Correlation strength:
0 – 0.29 = weak
0.30 – 0.59 = moderate
0.60 – 1 = strong
Simple Linear Regression
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Syntax
REGRESSION
/STATISTICS COEFF OUTS R ANOVA
/DEPENDENT SHTOT
/METHOD=ENTER (put in variables) age.
Method of least squares, trying to minimize squared error.
The Regression Equation is in the form:
Predicted outcome = y-axis intercept + slope x(age)
Predicted outcome = a + b (age)
From SPSS
Constant (B) = where lines hits the y axis
Is the regression significant
Use f formula
How useful is the data?
R2=Sums of squares regression/sum of squares total
How appropriate is it to do a regression analysis
Normality of the residual NOT dependent variable
Normality
 PPLOT
 Residuals should plot on a straight line to satisfy normality (does
not need to be perfect but no systematic deviation from line of
observed values)
Linearity and constant variances
 Scatter plot
 Equal number of residuals above and below “0” line as you move
across the graph
 Constant variance – no fanning but even band of residuals
Independence
 No participant measured twice
 No participants influenced other participants
Hat = predicted
Sigma = Std Deviation
Assumption of constant/conditional variance
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Assumes standard deviation is the same along the regression
line
The variance of our residual is the same all along the regression
line
Measured by standard deviation
Multiple Regression
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For ordinal variable we check that are there enough people on
each point are they dispersed enough. Do ordinal variables need
to downgraded or upgraded?
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N= number in sample
> = greater than
variable per predictor ratio, so for a research question with 2
predictors and a desired 10 variable per predictor you would
write
n>10x2
Univariate data description
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Produce graphical summaries (histogram, error bar plot)
Comment on distributions (5 points)
Summarise with appropriate numerical values
Write global summary statement of what you have found
Normally mostly interested on the dependant variable
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Better to do all three for project report
Other two independent variable see week 3 slides.
Summary of step 3
 The three variables were approx. normally distributed. The
dependent variable, Mental Impairment, ranged from 17 to 41
(mean, 27; standard deviation, 5.5). For the two predictors life
events ranged from 3 to 97 (44; 22.6) and parents years of
education ranged from 3 to 97 (57; 25.3).
Bivariate data description
 To what extent can we predict our dependent variable from our
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independent variable
Plot DV against each IV, Do not need to plot IV’s against each
other
Comment on the plots (7 points)
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Consider outliers
Write gobal summary statement of what has been found
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Syntax: graph/scatter(matrix)=MI life_events SES.
Scatterplots for mental impairment against life events and SES show a
positive and negative linear relationship respectively. Both relationships
appear of low-moderate strength and only low correlation. The graph
shows no unusual characteristics although there may be one outlier for
the relationship between MI and SES.
5. Produce Correlation Matrix
 Consider collinearity, Multicollinearity
 Consider statistical significance of DV correlation with each IV
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Consider correlations between IV’s
Write summary
Collinearity (multicollinearity) occurs when two or more IVs are so
correlated that one can be predicted (almost) exactly from one or more of
the others
Made possible by a combination of:
 Multiple IVs
 Non-orthogonality due to observational design
To occur requires strong correlations between IVs
Indicators
 Large correlations among IV’s
 Large changes in coefficients and/or Standard eror when a new
IV is added to the model
SPSS calculates tolerance and variance inflation factor (=1/tolerance)
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Tolerance regresses x on all other IVs and assesses how much
of the variance in X is not accounted for by the other IV’s
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VIF>10 is a concern
Means tolerance<0.1 IV has little to distinguish it from a
combination of other IVs
Summary of step 5
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Both predictors are statistically significantly correlated with the
dependent variable. Life events (r=0.37; p=0.018) while
socioeconomic status has a correlation of similar degree with
mental impairment (r= -0.40; p=0.011). The two predictors are
very weakly correlated (r=0.12; p=0.45) and thus give no
reason to expect collinearity.
Step 6 Does it fit full model
 Test full model significance
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Consider alpha level
Comment on R^2
Comment on unusual values
Construct summary table and comment
What is the full model? This depends on theory
 Full additive model: includes main effects of all IV’s
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Full model includes all main effects and interaction
For this example: additive model
Test assumptions
 Residuals are statistically independent: known to be true
because of sampling method.
 Residuals are normally distributed: check with normal probability
plot
 Residuals have constant variance (root-MSE) (homoscedacity):
check with plot of residuals (vertical) v predicted values
(horizontal)
 Linearity: check with plot of residuals (vertical) v IVs (horizontal)