Download Week 1: Entering Data And Checking Assumptions

PSYCH 706: STATS II
Class #1
WHAT IS THE POINT OF THIS CLASS?
• Reinforce the basics you learned in Stats I, focusing
on mostly parametric statistics
• Help you feel comfortable:
• Exploring data and looking for outliers
• Choosing the appropriate statistical test(s)
• Explaining results of tests clearly to audiences
• Displaying your results as clearly and simply as possible
SYLLABUS
• “Dr. Stewart” or “Jenny” – either is fine
• Office Hours in SB A-312: by appt.
• Grade based on:
• Four homework assignments (40% grade)
• Two exams in-class (open book) (30% grade)
• Final take-home exam (20% grade)
• Attendance / Participation (10% grade)
REQUIRED TEXTBOOK
CLASS STRUCTURE
• Tuesdays 5-7:50pm
• 5-6pm Lecture/Discussion about main
concepts
• Quick break, then 6-7pm SPSS tutorial
• 7-8pm lab, you can work on homework
and I’ll answer questions you might have
This picture makes me
laugh - Both men and
women can be good
at statistics, but
according to clipart
you wouldn’t think that
was true
BLACKBOARD SITE FOR THIS CLASS
Contains:
• SPSS files used in the Field chapters so you can
practice along with book
• Assignments
• Additional required readings
• SPSS handouts we’ll review during in-class tutorials
• PowerPoint slides
PSYCHOLOGICAL RESEARCH
• You want to understand relationships between
particular constructs
• Develop a model about how constructs are related in a
population
• Collect data on variables representing those constructs in a
sample of the population
• See how well your hypotheses about the population fit the
actual sample data you collected = STATISTICS
Depends on two factors: population mean (μ)
and standard deviation (σ)
NORMAL
DISTRIBUTION
We rarely have access to
actual population
parameters: means (μ) and
standard deviations (σ)
Instead we use sample
statistics to estimate means
and standard deviations
and compare them to what
we would expect in the
normal distribution
CHARACTERIZING NORMAL
DISTRIBUTION
• Review descriptive statistics
• Go over formulas for mean and standard deviation
• Purpose of z-scores
• Graphing mean differences: Confidence intervals vs. standard error
• Assumptions of normal distribution and how to check for violations
• Reducing bias (outliers, violations) in your data
DESCRIPTIVE STATISTICS
Center
Spread
Shape
Mode
Range
Skewness
Median
Variance
Kurtosis
Mean
Standard Deviation
Sum up all sample
scores and divide
by # scores
RANGE/QUARTILES (USED IN
BOXPLOTS)
• Range = Largest number minus smallest
• Quartiles = 3 values split sorted data into four equal parts.
• Second Quartile = median
• Lower quartile = median of lower half of the data
• Upper quartile = median of upper half of the data
• Interquartile Range
NOW WE’RE GOING TO
GO THROUGH STEP BY
STEP TO SHOW:
WHY THE STANDARD
DEVIATION IS USED TO
REFLECT
THE SPREAD OF A
DISTRIBUTION
DEVIANCE
We can calculate the spread of scores (the error in our
model) by looking at how different each score is from
the center of a distribution (the mean):
Score
Mean
Deviance
7
4
3
8
4
4
2
4
-2
0
4
-4
3
4
-1
TOTAL DEVIANCE:
0
DEVIANCE
PROBLEM: When we calculate total deviance, since
some deviances (errors) will be negative and others will
be positive, they’ll cancel each other out!
Score
Mean
Deviance
7
4
3
8
4
4
2
4
-2
0
4
-4
3
4
-1
TOTAL DEVIANCE:
0
SUM OF SQUARED ERRORS (SS)
SOLUTION: square each individual deviance and THEN
sum them up!
Score
Mean
Deviance
Deviance Squared
7
4
3
9
8
4
4
16
2
4
-2
4
0
4
-4
16
3
4
-1
1
SS: 46
SUM OF SQUARED ERRORS (SS)
PROBLEM: The larger the number of observations, the
larger SS will automatically be.
Score
Mean
Deviance
Deviance Squared
7
4
3
9
8
4
4
16
2
4
-2
4
0
4
-4
16
3
4
-1
1
SS: 46
VARIANCE (S²)
SOLUTION: Divide the SS by the number of observations
(minus 1*) to scale it!
* You lose 1 degree of freedom for estimating the mean in the first place
Score
Mean
Deviance
Deviance Squared
7
4
3
9
8
4
4
16
2
4
-2
4
0
4
-4
16
3
4
-1
1
SS: 46
N=5 observations
s² = 46/4 = 11.5
VARIANCE (S²)
PROBLEM: Then our measure is in units squared, which is
confusing to report in papers.
* You lose 1 degree of freedom for estimating the mean in the first place
Score
Mean
Deviance
Deviance Squared
7
4
3
9
8
4
4
16
2
4
-2
4
0
4
-4
16
3
4
-1
1
SS: 46
N=5 observations
s² = 46/4 = 11.5
STANDARD DEVIATION (S; OR SD)
SOLUTION: Take the square root of the whole darn
equation and then you’re back in regular units! Whew!
s = sqrt (11.5 units
squared) =
3.39 units
SAME MEAN, DIFFERENT
STANDARD DEVIATION
SKEWNESS
Symmetry
of
Distribution
Tail pointing at
high values
Tail pointing at
low values
KURTOSIS
Leptokurtic
Platykurtic
Heavy Tails
Light Tails
Heaviness
of the
Tails
SPSS DESCRIPTIVE STATISTICS:
ONE FORMAT
SPSS
DESCRIPTIVE
STATISTICS:
ANOTHER FORMAT
CREATING A STANDARD SCORE:
Z-SCORES
• Allows us to calculate probability of a
sample score occurring within normal
distribution
• Enables us to compare two scores that
are from different normal distributions
• Expresses a score in terms of how many
standard deviations it is away from the
mean
• z distribution: mean of 0 and SD = 1.
NORMAL DISTRIBUTION
CALCULATING Z-SCORES
Using population
parameters (we typically
don’t know these)
Using sample statistics
based on our own data set
Z-SCORE CUTOFFS
Top/Bottom of Distribution
Z-Score
2.5% (5% two-tailed)
+/- 1.96
.05% (1% two-tailed)
+/- 2.58
.005% (0.1% two-tailed)
+/- 3.29
Once you’ve calculated descriptive statistics,
what is a simple and clear way to display them?
GRAPHING MEAN DIFFERENCES
Confidence Intervals (CI)
Standard Error (SE)
Vs.
CI =
SE =
+/- 1SE
z = -1.96
z = +1.96
95% CI or +/- 2SE
CONFIDENCE INTERVALS (CI)
• True mean reaction time for all women and
men in the population is unknowable.
• 95% CI = If we repeatedly studied a different
random sample of women, 95% of the time
the true mean for all women will fall within
these upper/lower values. You can do the
same calculation for men.
• Say you did significance testing and decided
to plot means with CIs. CIs can overlap as
much as 25% and there can still be a
significant (p<.05) difference between means
for men and women.
STANDARD ERROR (SE)
• +/- 1 SE = 68% chance that the true mean falls
within this range (sort-of like 68% of a CI).
• +/- 2 SE = 95% chance that the true mean falls
within this range (almost equivalent to a CI).
Typically you only plot +/- 1 SE though.
• When SEs overlap, differences between men
and women are not significant at p<.05 (here
they are far from overlapping). Actually to be
significant, there needs to be about ½ an error
bar’s space between the two means for
significance to occur.
So which do you use, CIs or SEs? It often depends on the
preference of a particular journal as well as your personal
preference.
What about line versus bar graphs for plotting means?
Lines work well for continuous data
Bars work well for categorical data.
Often people go with personal preference.
ASSUMPTIONS OF THE NORMAL
DISTRIBUTION*
• Additivity and Linearity
• Normality
• Homoscedasticity /
Homogeneity of
variance
• Independence of errors
*As sample sizes for each group approach 30 or larger, the less you have to worry about this
because sample stats approximate population parameters exponentially as sample size
increases
ASSUMPTIONS OF THE NORMAL
DISTRIBUTION
• Additivity and Linearity:
Outcome of any model
we create is linearly
related to predictor
variables
ASSUMPTIONS OF THE NORMAL
DISTRIBUTION
• Normality
Confidence intervals,
sampling distribution of
means, and errors all
need to be normally
distributed
ASSUMPTIONS OF THE NORMAL
DISTRIBUTION
• Homoscedasticity /
Homogeneity of
variance
ASSUMPTIONS OF THE NORMAL
DISTRIBUTION
• Independence of errors
They should not be
correlated with each
other!
CHECKING FOR OUTLIERS AND
VIOLATION OF ASSUMPTIONS
• Histograms
• Boxplots
• Q-Q (quartile-quartile) plots
• P-P (probability-probability) plots
• Scatterplots
• Skewness/Kurtosis z-score checks
• K-S test and Levine’s test
HISTOGRAM
• Plots variable values on xaxis
• Plots frequency of responses
on y-axis
• Can eyeball skewness,
kurtosis, and outliers with
help from normal curve
outlined in black
BOXPLOT
• Uses info from the median and
interquartile range to
determine outliers and
extreme values
• Outliers = any scores in the
upper quartile +
[1.5*interquartile range]
• Extreme scores = any scores in
the upper quartile +
[3*interquartile range]
P-P AND Q-Q PLOTS
• PDF = probability distribution function –
plotting two non-cumulative datasets
against each other
Example: Q-Q plot
• CDF = cumulative distribution function –
plotting two cumulative (range: 0-1)
datasets against each other
Example: P-P plot
Q-Q PLOT
• Plots observed sample values
on the X-axis and the
expected values (assuming a
normal distribution) on the Yaxis.
• If the sample distribution is
distributed exactly like a
normal distribution, the points
should fall on a straight line.
• Plots quartiles in your sample
data, not all points
• Magnifies deviations from
proposed distribution on tails
INTERPRETING Q-Q PLOTS
P-P PLOT
• Plots cumulative probabilities, with
observed probabilities on the Xaxis and expected probabilities
given the normal curve on the Yaxis.
• Again, if the sample were exactly
normally distributed, the points
would lie on a straight line
• Plots all probabilities from your
data
• Magnifies deviations from normal
distribution in middle
SCATTERPLOT
• Plotting data for one
variable against
another variable
• Look for outliers
• Check for
homoscedasticity
SKEWNESS/KURTOSIS Z-SCORE
CHECKS
• Divide each
skewness or
kurtosis value by
its standard error
and compare to
a z-distribution
Exam Performance
• Skewness -.373/.238 = -1.56
• Kurtosis -.852/.472 = -1.81
Exam Anxiety
• Skewness -2.012 /.238 = -8.45
• Kurtosis 5.192/.472 = 11
WITHIN
NORMAL
LIMITS
EXTREMELY
NONNORMAL!
Top/Bottom of Distribution
Z-Score
2.5% (5% two-tailed)
+/- 1.96
.05% (1% two-tailed)
+/- 2.58
.005% (0.1% two-tailed)
+/- 3.29
KOLMOGOROV-SMIRNOV (K-S)TEST
• Testing normality assumption
• Compares scores in your sample to a normally
distributed set of scores with the same mean and
standard deviation
• If p < .05 your sample distribution is significantly
different from the normal distribution
• Shapiro-Wilks test does the same thing
LEVINE’S TEST
• Testing homogeneity of variance assumption
• The variance of your outcome variable
should be the same for each groups (e.g.,
depressed vs. non-depressed)
• Tests whether variances in each group are
equal
• If p < .05, variances are unequal
REDUCING BIAS
• Trim the data
• Winsorizing
• Analyze with robust methods
• Transform the data
REDUCING BIAS
• Trim the data: remove outliers based on a rule
• %-based rules = trimmed mean, M-estimator (trims your data for you)
• SD-based rules = certain # SDs above/below mean; but SD/mean influenced by
outliers in the first place so the criterion used to remove them is inherently biased
• Winsorizing: replacing outliers with next-highest score that is not an outlier
(e.g., a score 3 SD from the mean)
• Analyze with robust methods (non-parametric tests, see Chapter 6 Field; or
bootstrapping: sampling with replacement from our dataset 1000-2000x to
get estimated confidence interval and SE for our data)
• Transform the data (log, square root, reciprocal)
STEPS: EXPLORING YOUR DATA
• Let’s say you collected the same data in two groups (depressed and
non-depressed). You entered the data in SPSS. What next?
• You’re going to do the following, once across ALL subjects, and once
for each group separately:
 Descriptive statistics (mean, standard deviation, range, skewness, kurtosis
etc.)
 Graphs (histograms, boxplots, P-P plots, Q-Q plots)
 Check if your data is normally distributed (compute z-scores for skewness,
kurtosis, compute K-S and Wilks-Shapiro tests)
 Figure out how you want to handle outliers and extreme scores
• Compute Levine’s test to determine whether groups violate
homogeneity of variance assumptions
• Reduce bias (e.g., Perform transformations on any problematic
variables and test how well the distribution changes w/ Levene’s test)
SPSS TUTORIALS (HANDOUT)
AFTER THE BREAK!
DETRENDED Q-Q PLOT
Here, the Y-axis is the
deviation (difference)
between what was
observed and what was
expected. This plot
sometimes makes the
pattern easier to decipher
(note the clear “S” pattern
indicating skew)