Transcript Methods of Presenting and Interpreting Information

Methods of Presenting and
Interpreting Information
Class 9
Why Use Statistics?
• Describe or characterize information
• Allow researchers to make inferences on
samples vis-à-vis the population from
which it was drawn
• Assess validity of hypotheses
Descriptive Measures
• Basic Measures
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Central Tendency
• Means, median, mode
• Supports group comparisons
– Variability – whether the Score is Typical
• Variability is important because it tells us about whether
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the distributions of scores across groups are equivalent
Standard deviations is the primary measure (about two
thirds of scores tend to fall within one SD of the mean)
Association or correlation – whether variables go
together
• Positive or negative association
• Association tells us what happens to one variable when the
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other one changes
No causal claim
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Hypothesis Testing
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Evaluation of the null hypothesis is often stated in terms
of the differences in scores between groups. If they are
different, we then ask whether the differences were
observed by chance (ie, if we repeated the experiment,
would we observe the same result? Or a similar result?
How often would we observe these differences?)
Types of Error
Type I error – we falsely reject the null hypothesis when in fact it
may be true. That is, we assume differences that do not exist. we
assume that populations are different when they are alike vis-avis the IV-DV relationship
Type I errors are assessed by the level of significance we choose
in our statistical tests B if we set significance at .05, we assume
that the differences we observed did not happen by more than a
5% chance. That is, we are accepting the odds that the sample
differences might have appeared by chance fewer than 5 times in
one hundred. The more extreme the criterion, the less likely the
sample differences occurred by chance
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Type II error – we accept the null hypothesis when, in fact, it is
false. That is, we conclude that populations do not differ when
in fact they do
Type II errors occur in inverse proportions to Type I errors.
That is, we fail to recognize population differences when they
may exist.
The only way to reduce Type II error while maintaining
a high threshold for Type I error is to use larger sample
sizes, or use statistical methods that use more of the
available information
Statistical power of a test is the probability that the test
will reject the hypothesis tested when a specific
alternative hypothesis is true. To calculate the power of
a given test, it is necessary to specify β (the probability
that the test will lead to the rejection of the hypothesis
tested when that hypothesis is true) and to specify a
specific alternative hypothesis
http://www.stat.sc.edu/~ogden/javahtml/power/power.html
Statistical Tests
• Choosing a model
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Statistical procedures should be keyed to the level
of measurement in both the independent and
dependent variables
• Tests of Association
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Chi square -- differences between observed and
expected frequencies
• Tests of Differences between Means
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ANOVA models, ANCOVA models
• Tests of differences in distributions (means, std deviations)
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between groups
Controls for covariates in instances where you assume
there may be differences between groups that are
systematic (not random)
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Multivariate Models
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Ordinary Least Squares (OLS) Regression, or Multiple
Regression
tells you which combination of variables, and in what priority,
influence the distribution of a dependent variable.
It should be used with ratio or interval variables, although there is
a controversy regarding its validity when used with ordinal-level
variables.
OLS regression is used more often in survey research and
non-experimental research, although it can be used to
isolate a specific variable whose influence you want to test
You can introduce interaction terms that isolate the effects
to specific subgroups (eg, race by gender).
If you do it right, you can control and eliminate statistical
correlations between the independent variables
Logistic Regression is a form of regression specifically
designed for binary dependent variables (e.g., group
membership)
Fitting a Regression Line
Fitting a Regression Line, from Eisenberg and Wells (2006)
Example of OLS Regression Results, from Fagan and Tyler (2005)
Complex Causation
• Structural Equations Models
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This family of statistical procedures looks at complex
relationships among multiple dependent variables
over time. It can accommodate feedback loops, or
hypothesized reciprocal relationships. It gives you a
probability estimate for each path
• Hierarchical Models
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When effects are nested – when independent
variables exist at more than one ‘level’ of explanation
– School research example
• School factors (e.g., teacher experience, average SES of
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parents, computer resources, parent involvement)
Individual factors (e.g., student IQ, family income, parental
supervision, number of siblings, sibling school performance)
Example of SEM Analysis, from Tyler and Fagan (2006)
Legitimacywave 1
.31
Gender
Ethnicity
.23
-.31
Age -.47
Fairness of 
treatment
.42
.33
.23
Fairness of
decision
making
64%
Outcome
fairness
Education
.19
Performancewave 1
Legitimacywave 2
.32
.65
Income
57%
.56
Cooperation–
wave 1
.24
.47
.47
Cooperationwave 2
Performance
-wave 2
50%
Figure 3. Cooperation with the police (n = 255; CFI = 0.88)
How Good is the Model?
What Does It Tell Us?
• Most multivariate models generate
probability estimates for each variable in
the model, and also for the overall model
– Model Statistics: “model fit” or “explained
variance” are the two most important
– Independent Variables
• Coefficient estimate
• Standard Error
• Statistical Significance
• Alternatives to Statistical Significance
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Odds Ratio – the odds of having been exposed
given the presence of a disease (ratio) compared to
the odds of not having been exposed given the
presence of the disease (ratio)
Risk Ratio – the risk of a disease in the population
given exposure (ratio) compared to the risk of a
disease given no exposure (ratio, or the base rate)
Attributable Risk –
(Rate of disease among the unexposed – Rate of disease among the exposed)
(Rate of disease among the exposed)
Example from
Breast-Feeding Study
• In this example, the odds ratio is a way of comparing
whether the probability of a certain event is the same for
two groups. An odds ratio of 1 implies that the event is
equally likely in both groups. An odds ratio greater than
one implies that the event is more likely in the first
group. An odds ratio less than one implies that the event
is less likely in the first group.In the control row, the
odds for exclusive breast feeding at discharge are 27 to
20 or odds slightly in favor of exclusive breast feeding.
• In the treatment row, the odds are 10 to 32 or odds
strongly against exclusive breast feeding.
• The odds ratio for exclusive breast feeding at discharge
is (27/20) / (10/32) = 4.32.
• Since this number is larger than 1, we conclude that
breast feeding at discharge is less likely in the
experimental group.