# A Statistical Sampler

Document Sample

```					  A Statistical Sampler

Dr. Ed Greenberg
ASU College of Nursing
Center for Research & Scholarship

August 2005

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To understand God's thoughts we must study
statistics, for these are the measure of His
purpose.

— Florence Nightingale

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Statistical Terms Crossword

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To behold is to look beyond the fact; to observe, to
go beyond the observation. Look at the world of
people, and you will be overwhelmed by what you
see. But select from that mass of humanity a well-
chosen few, and observe them with insight, and
they will tell you more than all the multitudes
together.

— Paul D. Leedy
From his book, ―Practical Research,‖ 1993

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Choosing the Appropriate Statistic

Some factors to consider:
 Research design
 Number of groups
 Number of variables
 Level of measurement
(nominal, ordinal, interval/ratio)

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Statistical Methods

Statistical Methods

Descriptive Methods                  Inferential Methods

Univariate                           Applied to means

Bivariate                    Applied to other statistics

Multivariate

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Descriptive Statistics

Descriptive Methods

Univariate            Bivariate                 Multivariate

shape                correlation            multiple regression

spread               regression

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Inferential Statistics

Inferential Methods

Applied to means             Applied to other statistics

2 groups: t-test

>2 groups: ANOVA

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While the individual man is an insoluble puzzle, in the
aggregate he becomes a mathematical certainty. You
can, for example, never foretell what any one man will
be up to, but you can say with precision what an
average number will be up to. Individuals vary, but
percentages remain constant. So says the statistician.

— Arthur Conan Doyle

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Some Statistics-Related Web Sites

The University of Kansas Virtual Statistical Assistant
http://www.ku.edu/~coms/virtual_assistant/vsa/

Biostatistics for the Clinician Hypertext Glossary
Part 1:
http://www.uth.tmc.edu/uth_orgs/educ_dev/oser/LGLOS1_0.HTM
Part 2:
http://www.uth.tmc.edu/uth_orgs/educ_dev/oser/LGLOS2_0.HTM

Research Methods Knowledge Base
http://www.socialresearchmethods.net/kb/

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Types of Statistics

   Descriptive statistics characterize the attributes of
a set of measurements. Used to summarize data, to
explore patterns of variation, and describe changes
over time.

   Inferential statistics are designed to allow
inference from a statistic measured on sample of
cases to a population parameter. Used to test
hypotheses about the population as a whole.

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Requisite Conditions for Causation

In order for X to cause Y:

 X & Y must be associated

 X must precede Y in time

 X contains unique information about Y that is
not articulated elsewhere

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The invalid assumption that correlation implies
cause is probably among the two or three most
serious and common errors of human reasoning.

— Stephen Jay Gould, The Mismeasure of Man

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Smoking is one of the leading causes of statistics.

— Fletcher Knebel

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Randomization

 Random selection is how you draw the
sample for your study from a population.
 This is related to the external validity, or
generalizability, of your results.

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Randomization

 Random assignment is how you assign your
sample to groups or treatments in your study.
 This is related to internal validity.
 Random assignment is a required feature of a
true experimental design.

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Randomization

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Variables

 Variables are qualities, properties, or
characteristics of persons, things, or situations
that change or vary and are manipulated,
measured, or controlled in research.

 More simply stated:
Variables are things that we measure, control,
or manipulate in research.

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Types of Variables

 Independent variables are manipulated or
varied by the researcher, for example,
intervention or treatment.
 Dependent variables are the responses,
outcomes, etc. that are measured by the
researcher.
 Extraneous variables are not part of the
research design, but may have an impact on
the dependent variable(s).

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Levels of Measurement

 Nominal

 Ordinal

 Interval

 Ratio

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Nominal-Level Variables

 Data are organized into categories
 Categories have no inherent order
 Categories are exclusive
 Categories are exhaustive
 Examples are sex, ethnicity, marital status

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Examples of Nominal-Level Questions

 Do you have a loss of appetite?

 Do you smoke a lot?

 What is your ethnicity?

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Ordinal-Level Variables

 Categories can be ranked in order
 Intervals between categories may not be
equal
 Examples are socioeconomic status, level of
education attained (elementary school, high
school, college degree, graduate degree)

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Examples of Ordinal-Level Questions
   Would Intervention X be your 1st, 2nd, or 3rd choice of
treatment for Condition Y?
1   First choice
2   Second choice
3   Third choice

   Beck Depression Scale – Sadness Item
0   I do not feel sad
1   I feel sad
2   I am sad all the time and I can’t snap out of it
3   I am so sad or unhappy that I can’t stand it

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Interval-Level Variables

 Distances between levels of the scale are
equal

 Assumed to be a continuum of values

 An example is temperature (measured in
Fahrenheit or Centigrade)

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Examples of Interval-Level Variables

 IQ scores

 GRE scores

 Composite scores of multi-item scales

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Ratio-Level Variables

 Equal spacing between intervals
 Have an identifiable absolute zero point
 Examples are weight, length, volume, and
temperature (measured in Kelvin)
 In statistical analysis, typically there is no
distinction made between interval level and
ratio level

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Same Variable, Different
Levels of Measurement

Interval level:
What is your age in years?   ____

Ordinal level:
What is your age group?
 18 years or younger
 19-44 years
 45 years or older

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Importance of Levels of Measurement

 Level of measurement is associated with the
type of statistical method used.
 Higher levels of measurement provide more
information than do lower levels.
 In general, you should use the highest level
of measurement possible. For example,
measure actual age in years, not in age
groups.

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Some Major Types of Analyses

 Description

 Relationships among variables

 Differences between groups or treatments

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There are three kinds of lies –
lies, damned lies and statistics.

— Benjamin Disraeli

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Measures of Central Tendency
Level of
Statistic
Measurement

Nominal          Mode        What is the most frequent value?

What is the middle score?
Ordinal          Median
(50% above and 50% below)

What is the average?
Interval/Ratio   Mean        (Sum of all scores divided by the
number of scores)

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Example of Central Tendency

15,20,21,20,36,15,25,15

15,15,15,20,20,21,25,36

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Example of Mode

RACE Race of Respondent                        Race of Respondent
1400
Frequency   Percent
1 white       1257        83.8           1200

2 black        168        11.2
1000
3 other          75        5.0
Total         1500      100.0
800

Statistics                          600

RACE Race of Respondent                    400
N     Valid        1500
Frequency

Missing         0                    200
Mode                  1
0
w hite        black    other

Race of Respondent

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Example of Median

EDUC Education level
10
Cumulative
Frequency   Percent     Percent
4 Some high school               1         4.2           4.2   9
5 Completed high school          6        25.0         29.2
6 Some college                   6        25.0         54.2
8
7 Completed college              3        12.5         66.7
8 Some graduate work             4        16.7         83.3
9 A graduate degree              4        16.7        100.0    7
Total                           24      100.0

6
Statistics
5
EDUC Education level
N      Valid                  24
Missing                 0                                4

Median                      6.00
3
N=                24

Education level

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Example of Mean

Age of Respondent
200
MEAN

100

Std. Dev = 17.42
Mean = 46
0                                                   N = 1495.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Age of Respondent

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I abhor averages. I like the individual case. A man
may have six meals one day and none the next,
making an average of three meals per day, but that
is not a good way to live.

— Louis D. Brandeis

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Measures of Variation

Level of
Statistic
Measurement

How many different values are
Nominal           Number of categories
there?

What are the highest and lowest
Ordinal           Range
values?

What is the average deviation
Interval/Ratio    Standard Deviation
from the mean?

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Curves of Distribution

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Normal Distribution

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Normal Curve

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Example: Number of categories

RACE Race of Respondent                         Race of Respondent
1400
Frequency   Percent
1 white       1257        83.8           1200

2 black        168        11.2
1000
3 other          75        5.0
Total         1500      100.0
800

600

400
Frequency

200

0
w hite        black    other

Race of Respondent

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Example of Range
EDUC Education level

Cumulative
Frequency   Percent     Percent      10
4 Some high school             1         4.2           4.2
5 Completed high school        6        25.0         29.2
6 Some college                 6        25.0         54.2    9
7 Completed college            3        12.5         66.7
8 Some graduate work           4        16.7         83.3
8
9 A graduate degree            4        16.7        100.0
Total                         24      100.0
7

Statistics
6
EDUC Education level
N        Valid                  24
5
Missing                 0
Median                        6.00
Range                            5                            4

Minimum                          4
Maximum                                                       3
9                             N=                 24

Education level

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Example of Standard Deviation

Age of Respondent
200
-1 SD           MEAN               +1 SD

100
Frequency

Std. Dev = 17.42
Mean = 46
0                                                                               N = 1495.00
20        30        40        50        60        70        80        90
25        35        45        55        65        75        85

Age of Respondent

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Measures of Relationships

Level of
Statistic
Measurement

Nominal              Phi statistic ()

Ordinal              Spearman rho () correlation

Interval/Ratio       Pearson correlation (r)

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Statistics have shown that mortality increases
perceptibly in the military during wartime.

— Robert Boynton

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Example of Spearman Correlation
RINCOM91 Respondent's Income                                                       DEGREE RS Highest Degree

Frequency   Percent     Valid Percent                                           Frequency   Percent    Valid Percent
Valid     1 LT \$1000               26        1.7              2.7            Valid       0 Less than HS          279        18.6           18.6
2 \$1000-2999             36        2.4              3.8                        1 High school           780        52.0           52.1
3 \$3000-3999             30        2.0              3.2                        2 Junior college          90        6.0             6.0
4 \$4000-4999             24        1.6              2.5                        3 Bachelor              234        15.6           15.6
5 \$5000-5999             23        1.5              2.4                        4 Graduate              113         7.5             7.6
.                         .           .               .                        Total                  1496        99.7          100.0
.                         .           .               .            Missing                                4         .3
.                         .           .               .                        Total                      4         .3
19 \$50000-59999          38        2.5              4.0            Total                              1500      100.0
20 \$60000-74999          23        1.5              2.4
21 \$75000+               44        2.9              4.6
Total                  947        63.1           100.0
Missing                          553        36.9
Total                           1500      100.0

Correlations

RINCOM91
Respondent's
Income
Spearman's rho     EDUC Highest Year Correlation Coefficient        .363**
of School Completed Sig. (2-tailed)              .000
N                             945
**. Correlation is significant at the .01 level (2-tailed).

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Scatterplot of Self Esteem By Height

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Relationship Between Two Variables

Positive      Negative                Curvilinear
Correlation   Correlation              Relationship

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Example of Pearson Correlation
     Variable HEIGHT is measured in inches
     Variable ESTEEM is the average of 5 items
measured on a four-point scale (1-4)
Statistics                             4.0

HEIGHT    ESTEEM
N                 Valid                  24         24
Missing                 0          0        3.5
Mean                                66.7917     2.7583
Std. Deviation                      7.03395     .59558
3.0

Correlations
2.5
ESTEEM
HEIGHT    Pearson Correlation          .347
Sig. (2-tailed)              .097
N                              24                   2.0
ESTEEM

1.5
50          60      70      80      90

HEIGHT

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Example of Chi-Square Test

RACE * SEX Crosstabulation

SEX
1 Male                     2 Female                        Total
Count   % within SEX        Count    % within SEX         Count    % within SEX
RACE    1 white      552        86.1%            705         82.1%            1257         83.8%
2 black       66        10.3%            102         11.9%             168         11.2%
3 other       23          3.6%            52           6.1%             75           5.0%
Total                641       100.0%            859        100.0%            1500        100.0%

Chi-Square Tests

Asymp. Sig.
Value           df         (2-sided)
Pearson Chi-Square       5.994 a             2           .050
N of Valid Cases          1500
a. 0 cells (.0%) have expected count less than 5. The
minimum expected count is 32.05.

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A Statistical Sampler

Take a 15 minute break!

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Statistical thinking will one day be as
necessary a qualification for efficient
citizenship as the ability to read and write.

— H.G. Wells

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Some Terminology

 Descriptive statistics
Statistics that allow the researcher to organize or
summarize data to give meaning or facilitate insight.

 Inferential statistics
Methods that allow inferences to be made from a
sample to a population

 Hypothesis testing
A statistical test of an expected relationship between
two or more variables

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Statistical inference

Statistical inference is the process of estimating
population parameters from sample statistics.

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Statistical inference may be used to ascertain
whether differences exist between groups...

90
80
70
Height in inches

60
50
40
30
20
10

Males              Females

Are males taller than females?

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... or whether there is a
relationship among variables.
4.0
SELF ESTEEM SCORE

3.5

3.0

2.5

GENDER
2.0
FEMALES
1.5                            MALES
20   30   40    50     60

AGE

Is there a relationship between age and self-esteem?
Does this relationship differ for males and females?

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Examples of Some Commonly Used Statistical Tests

Level of Measurement

Number of groups              Nominal           Ordinal            Interval/Ratio
t-test of sample
Kolmogorov-Smirnoff 1
1 group                 2   test                                 mean vs. known
sample test
population value
2 test                                   Independent
2 independent groups                      Mann-Whitney U test
samples t-test

2 dependent groups      McNemar test      Wilcoxon test           Paired t-test

>2 independent groups   2 test           Kruskal-Wallis ANOVA    ANOVA

Friedman ANOVA by       Repeated
>2 dependent groups     Cochran Q test
ranks                   measures ANOVA

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Some Commonly-Used Multivariate Methods

 Analysis of Variance and Covariance
Tests for differences in group means
 Multiple Regression Analysis
Estimates the value of a dependent variable
based on the value of several independent
variables

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Some Commonly-Used Multivariate Methods

 Reliability analysis
Assesses the consistency of multi-item scales
 Factor Analysis
Examines the relationships among variables and
reveals related sets of variables (constructs)
 Structural Equation Modeling
Methods for testing theories about the
relationships among variables

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Hypothesis Testing Decision Chart

Null Hypothesis (H0 ) is     Alternative Hypothesis
Reality
true                      (H1) is true
Decision

Type I error               Correct decision
Reject (H0 )
()                    (Power = 1 - )

typically .05 or .01            typically .80

Correct decision               Type II error
Don’t reject (H0 )                (1 - )                       ()

typically .95 or .99            typically .20

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Difference between two group means:
The independent samples t-test
Males and females are asked a question that is
measured on a five-point Likert scale:
To what extent do you feel that regular exercise contributes
to your overall health?

1   Strongly agree
2   Agree
3   Neither agree nor disagree
4   Disagree
5   Strongly disagree

Do males and females differ in their response to this question?

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25 males and 25 females answered our
question. Here is how they responded:

males

females

1        2      3     4   5
meanmales=2.5

meanfemales=3.2

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We can use the SPSS statistical package to run
an independent samples t-test:

First we enter the   Then we invoke the Independent
data into SPSS.      Samples T-Test procedure.

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We tell SPSS which is the dependent
variable and which is the independent
variable to use in performing the t-test:

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SPSS gives us summary statistics for each group:

Group Statistics

Std. Error
GENDER        N       Mean Std. Deviation          Mean
EXERCISE   1 male         25      2.56        1.158              .232
2 female       25      3.24        1.012              .202

The t-test reveals a significant difference between
males & females:
Independent Samples Test

t-test for Equality of Means
Mean
t         df        Sig. (2-tailed)   Difference
EXERCISE      -2.212            48             .032           -.68

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Reporting Results

 See the guidelines in the
APA Publication Manual,
Fifth Edition
 The manual provides very
specific instructions for
presenting statistical results.
 Example:
The mean exercise score for
females, 3.24, was
significantly higher than for
males, 2.56, t(48) = 2.12, p =
.032.

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Do the educational levels of males and females differ?
10

9
Education level

8
9   A graduate degree
8   Some graduate work        7
7   Completed college
6   Some college              6
5   Completed high school
4   Some high school          5
3   Completed grade school
2   Some grade school         4
1   No formal education
3
N=     14               10
Female            Male

Gender

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Because the dependent variable (education level)
is ordinal-level, we use the Mann-Whitney U Test.
Ranks

For each group, the                            GENDER            N           Mean Rank   Sum of Ranks
EDUC Education level 1 Female           14             13.46        188.50
Sum and mean of ranks
2 Male             10             11.15        111.50
Is computed.                                   Total              24

Test Statisticsb

The test statistics                                          EDUC
suggest that males’ and                                     Education
level
females’ education                Mann-Whitney U               56.500
Wilcoxon W                  111.500
levels do not differ in           Z                              -.807
this population.                  Asymp. Sig. (2-tailed)          .420
Exact Sig. [2*(1-tailed                a
.437
Sig.)]
a. Not corrected for ties.
b. Grouping Variable: GENDER

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Difference between two groups over time:
Repeated measures analysis of variance

 Asthmatic elementary school children are given
training intended to reduce the number of asthmatic
episodes.
 A control group is not given the training.
 Children’s school attendance is monitored during the
month before training is given to the intervention
group, and during each of the two months following
the intervention.

 Does the asthma training intervention improve the
school attendance relative to the control group?

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The experimental design:

Month 0    Intervention    Month 1     Month 2

Intervention Group         O            X             O           O

Control Group              O                          O           O

O = observation   X = treatment/intervention

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We can use the SPSS statistical package to
perform a repeated measures ANOVA on the
sample data:

First we enter the    Then we request the General
data into SPSS.       Linear Models procedure for
Repeated Measures.

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Here are the results involving time:
Tests of Within-Subj ects Effects

Measure: ATTEND

Type III Sum
Source             of Squares       df        Mean Square     F        Sig.
TIME                       .034           2          .017   1.695      .205     The time x group
TIME * GROUP               .080           2          .040   3.956      .033     interaction is
Error(TIME)                .244          24          .010                       significant.

And here are the results involving group:
Tests of Between-Subj ects Effects

Measure: ATTEND
Transformed Variable: Average
Type III Sum
Source       of Squares     df    Mean Square          F        Sig.
Intercept        28.271       1        28.271       6293.102    .000          The main effect
GROUP                .068     1          .068         15.201    .002          involving group
Error                .054    12          .004                                 is significant.

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This is a plot of the group means over time
Estimated Marginal Means of ATTEND

100%
Attendance (% of days)

90%

Intervention

Control
80%

70%
Month 0               Month 1             Month 2

TIME

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Factor Analysis Example

The General Social Survey (GSS) is an ―almost annual‖ personal
interview survey of U.S. households conducted by the National
Opinion Research Center.

In the 1993 GSS, approximately 1500 adult respondents (18
years or older) were asked about their music preferences.

Just for the fun of it, I performed a factor analysis on the music
questions to see if we could identify a pattern of underlying
dimensions, or factors, in the data.

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MUSIC GENRES
I'm going to read you a list
of some types of music.            Big Band              Folk
Bluegrass             Jazz
Can you tell me which of           Country/Western       Opera
the statements on this card        Blues or R & B        Rap
comes closest to your              Broadway Musicals     Heavy Metal
feeling about each type of         Classical
music. (HAND CARD ―B‖
TO RESPONDENT.)
RESPONSE CARD “B”
Let's start with big band
music. Do you like it very              1    Like Very Much
2    Like It
much, like it, have mixed
3    Mixed Feelings
feelings, dislike it, dislike it        4    Dislike It
very much, or is this a type            5    Dislike Very Much
of music that you don't                 8    DK Much About It
know much about?                        9    NA

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Factor Analysis Results
The factor analysis revealed four factors in the music preference items.
The varieties of music were associated with the factors as shown below:

Pattern Matrixa

Factor
1        2       3       4
CLASSICL Classical Music              .844    -.033   -.127    .054
OPERA Opera                           .715    -.004   -.032    .086
MUSICALS Broadway Musicals            .663     .109   -.024   -.104
FOLK Folk Music                       .502    -.064    .341   -.005
BIGBAND Bigband Music                 .459     .240    .125   -.171
JAZZ Jazz Music                       .035     .766   -.110    .029
BLUES Blues or R & B Music           -.024     .714    .106    .057
BLUGRASS Bluegrass Music              .070     .084    .753    .052
COUNTRY Country Western Music        -.084    -.034    .596   -.033
HVYMETAL Heavy Metal Music           -.012    -.016    .020    .602
RAP Rap Music                         .030     .074   -.004    .559
Extraction Method: Principal Axis Factoring.
Rotation Method: Oblimin with Kaiser Normalization.
a. Rotation converged in 8 iterations.

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Factor Analysis Results
FACTORS

F1                   F2                       F3                 F4

Classical              Folk

Musicals    Big Band

Heavy
Opera             Jazz        Blues   Bluegrass    Country                Rap
Metal

MEASURED VARIABLES

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Do not put faith in what statistics say until you
have carefully considered what they do not say.

— William W. Watt

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More Cool Statistics Web Sites

Rice Virtual Lab in Statistics
http://www.ruf.rice.edu/~lane/rvls.html

Multimedia Resources for Statistics Students
http://research.ed.asu.edu/msms/multimedia/multimedia.cfm

Statistics and Statistical Graphics Resources
http://www.math.yorku.ca/SCS/StatResource.html

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Without data, all you are is just another
person with an opinion.

— Unknown

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Statistical Power Analysis

 Prior to conducting a study, it is advisable
to conduct a statistical power analysis.

 Power is the probability that a statistical
test will detect a significant effect that
exists.

 The power analysis will suggest an
adequate sample size for the study.

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Four parameters related to
the power of a test:

 Significance level ()

    Sample size (n)

    Effect size (ES)

    Power (1 - )

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Relationship between power
and other parameters:

 As significance level () decreases
numerically, power decreases

 As effect size increases, power
increases

 As sample size increases, power
increases

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Conventions commonly used:

 Significance level ():                                  .05 *
.01
.001
 Effect size:                                      ―small‖
―medium‖ *
―large‖
 Power:                                                   .80 *
.90
*   Typical values for social/behavioral/health sciences

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Examples of Effect Size:

EFFECT SIZE
TYPE OF         MEASURE OF
TEST            EFFECT SIZE
SMALL      MEDIUM     LARGE

Independent       |mA-mB|
Samples T-                     .2           .5       .8
test                 

Product
Moment              rXY        .10         .30       .50
Correlation

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Testing a mean against a true alternative:
1 slightly larger than 0 (―small effect‖)

Sampling distribution of
means when H0 is true
Area=                                               Sampling distribution of
means when H1 is true

Area=1-

Area=

0   1

Region of nonrejection                         Region of rejection
Critical value

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Testing a mean against a true alternative:
1 quite a bit larger than 0 (―large effect‖)

Area=                                                             Area=1-

Area=

0                    1

Region of nonrejection                              Region of rejection
Critical value

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Relationship Between Alpha(),
Sample Size (n), and Power (1-)
Two group t-test of equal means (equal n's)
α             δ
Æ= 0.025 ( 2) Ê = 0.500
α             δ
Æ= 0.050 ( 2) Ê = 0.500
α             δ
Æ= 0.100 ( 2) Ê = 0.500
100

90

80
Power

power=.80

70

60

n=51        n=64     n=78
50
20               40           60             80                 100            120
Sample Size per Group

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The Power Analysis ―Bible‖

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There are a lot of statistical power analysis
resources (including interactive ―power
calculators‖) on the World Wide Web.

For example, see the StatPages.net web site at:
http://members.aol.com/johnp71/javastat.html#Power

Or, using a WWW search engine like Yahoo or Google,
use the search string:
statistical power analysis

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Getting Help

 For course assignments involving statistics,
see your instructor or teaching assistant.
 For help related to a masters thesis or applied
project, see your faculty advisor.
 Your instructor or advisor may confer with or
make an appointment as needed with a
statistician in the College of Nursing Center for
Research and Scholarship.

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Getting Help
The Statistics Hotline is
sponsored by a joint effort of the
ASU Committee on Statistics,
the Department of Mathematics
and Statistics, and the Division
of Graduate Studies. Its
services are available to
anyone affiliated with ASU and
needs assistance with their
ASU-related research.

http://www.asu.edu/graduate/statistics/hotline/

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An approximate answer to the right question
is worth a great deal more than a precise
answer to the wrong question.

— The first golden rule of mathematics,
sometimes attributed to John Tukey

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Statistical Terms Crossword Solution

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On the Web

This presentation is available online in Microsoft
PowerPoint format at:

http://www.public.asu.edu/~eagle/stat_sampler.ppt

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