Basic Descriptive and Inferential Statistics

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Basic Descriptive and Inferential Statistics

• Analytical Techniques for Public Service
• The Evergreen State College
• Winter 2010

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Where are we?You Have

• Identified your problem/research question
• Described why the issue is worth studying
• Conducted a literature review to see what others
  have done and to shed more light on your
  question
• Identified and operationalized your measures
• Identified a research design that is capable of
  answering to some degree your research question
• You will soon be in the field collecting your
  data
• Now What??

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Preparing Your Data for Analysis

• Prepare code categories (e.g., 1 female 2
  male)
• Prepare a codebook (this tells you the location
  of data and the meaning of the codes in a data
  file).
• Create a data file based upon the codebook (e.g.
  Excel, SPSS, SAS, JMP8, ASCII).
• Once the data are entered and cleaned and made
  analysis ready we are ready for analysis.

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Valuable Books for Your Arsenal

• Stevens, J. (2002) Applied Multivariate
  Statistics for the Social Sciences. Lawrence
  Erlbaum Associates 4th Ed.
• Blalock, H. (1979) Social Statistics McGraw Hill
  Revised 2nd Ed.
• Kraemer, H and Thiemann, S. (1987) How Many
  Subjects Statistical Power Analysis in Research.
  Sage Publications.
• Babbie, E. (2009). The Practice of Social
  Research. Wadsworth Publishing 12th Edition.

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What will be Considered

• Descriptive vs Inferential Statistics
• Basic Terminology
• Levels of measurement
• Strength of Association
• Hypothesis testing
• Type I and Type II Errors and Statistical Power

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Subject Matter of Statistics

• Descriptive Statistics - Tools and issues
  involved in describing collections of statistical
  observations, whether they are samples or total
  populations.
• Inferential Statistics (inductive statistics) -
  Deals with the logic and procedures for
  evaluating risks of inference from descriptions
  of samples to descriptions of populations (finite
  or infinite).

Loether, H and McTavish, D. (1976) Descriptive
and Inferential Statistics
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Basic Terms

• Variable (an attribute of a person or object that
  can take on different values).
• Distribution of a variable.
• Continuous or discrete variables
• Central tendency
• Range
• Dispersion/confidence intervals
• Levels of Analysis

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Measures of Central Tendency

• Central tendency
• The 3-Ms Mode, Median, Mode.
• Mode most frequent response.
• Median mid-point of the distribution
• Mean arithmetic average.

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Standard Deviation

• Normal Distribution Bell-shaped curve
• 68.26 of the variation is within 1 standard
  deviation of the mean
• 95 of the variation is within 1.96 standard
  deviations of the mean

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Applying the Standard Deviation

• Average test score 60.
• The standard deviation is 10.
• Therefore, 95 of the scores are between 40 and
  80.
• Calculation
• 602080 60-2040.

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ExerciseConfidence Intervals
N Range Min Max
Age 825 11.93 9.49 21.43
Mean SE Mean SE Std. Deviation Variance
14.5548 .06925 1.98898 3.956
Calculate and interpret a 95 confidence interval
for these data.
-
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Variable types

• Continuous variable Attributes are a steady
  progression (income, age). No gaps.
• Discrete variable Attributes are separated
  gappiness (gender, religious affiliation, race)

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Analysis

• Univariate Analysis Single Variable
• Bivariate Analysis Analysis of two variable
  simultaneously
• Multivariate Analysis Analysis of simultaneous
  relationship of several variables.

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Level of Measurement

• Nominal Data Categorical (e.g., gender, race)
• Ordinal Data Nominal More/less than (e.g.,
  social class, religiosity)
• Interval Data Nominal Ordinal How much
  more/less than. Categories have a standard unit
  of measure (e.g., Fahrenheit).
• Ratio Data Nominal Ordinal Interval a true
  zero (e.g., age, height).

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Levels of Measurement and Statistics Tests
1st Variable
2nd Variable-?
?
Single Variable Dichotomy Nominal Ordinal Interval/Ratio
Dichotomy Proportions, Percentages, ratios Diff of proportions, Chi Square Taub
Nominal (r cat) Proportions, Percentages, ratios Chi Square, Contingency C Phi/Cramers V Taub Yules Q Chi Square, Contingency C Cramers V Taub Yules Q
Ordinal Medians, quartiles, deciles, q deviations Mann-Whitney, runs, signed ranks Anova with ranks Gamma, Rank order corr, Kendallss Tau
Interval/ Ratio Means, Medians, SD Diff of Means Anova, Eta2, intraclass correlations Correlation and Regression
Blalock, Social Statistics
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Teen Pregnancy Risk Factors
YDP Program Participants (n) Healthy Youth Survey (n)
Participants getting mostly Ds Fs in school Grade 8 21.9 (137) 9.6 (7,923)
Participants getting mostly Ds Fs in school Grade 10 23.1 (78) 8.8 (7,673)
Participants getting mostly Ds Fs in school Grade 12 25 (56) 5.3 (5,684)
Participants who reported their mother did not finish high school Grade 8 27.7 (94) 7.3 (7,938)
Participants who reported their mother did not finish high school Grade 10 36.6 (71) 8.6 (7,688)
Participants who reported their mother did not finish high school Grade 12 23.1 (52) 9 (5,695)
Participants who used alcohol in the 30 days before pre-test Grade 8 31.7 (145) 18 (8,223)
Participants who used alcohol in the 30 days before pre-test Grade 10 45.2 (84) 32.6 (7,860)
Participants who used alcohol in the 30 days before pre-test Grade 12 53.4 (58) 42.6 (5,795)
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Measures of Association

• A class of statistical tests that are used to
  show the magnitude or strength of a relationship
  between variables.
• Significance tests are used to establish whether
  or not a relationship exists, and measures of
  association show the size of the relationship
  (weak, moderate, strong).
• Some also show the direction of the relationship
  ( for ordinal and interval-ratio variables).

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Measures of Association for Cross Tabulations
(examples)

• Lambda The strength of a relationship between
  two nominal variables.
• Phi The strength of a relationship between two
  dichotomous variables.
• Gamma The strength of a relationship between two
  ordinal variables.
• Values Range between 0 and or 1.
• Negative and positive values show the direction
  of the relationship, where applicable.
• The closer the value is to one, the stronger the
  relationship.

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Proportional Reduction of Error (PRE)

• PRE Proportional Reduction of Error The concept
  underlying these tests where
• The errors of prediction made when the
  independent variables is ignored (E1) and the
  errors of prediction made when the prediction is
  based on the independent variable (E2) are taken
  into account.
• If you know information about one variable, to
  what extent will that data help you predict
  information about another variable?

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General PRE formula

• of errors not knowing ind var (minus)
• of errors knowing ind var
• --------------------------------------------------
  --------------
• of errors not knowing ind var

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Are homeless people reporting mental health
problems more likely to request case management
than those who dont?
Exercise Calculate Lambda and Interpret
No Mental Health Problems Mental Health Problems TOTAL
Does Not Want Case Mgt 355 (69) 293 (45) 648
Wants Case Mgt 157 (31) 359 (55) 516
Total 512 (100) 652 (100) 1164
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Reading Tables

• Independent Variable Mental health problems
• Dependent variable Wants Case management
• Are those that have mental health problems more
  likely to say they will want case management?
• For each category of the independent variable,
  what is the percent distributions across the
  dependent variable?
• Percent distribution down columns

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Lambda

• An asymmetrical measure of association the value
  varies depending on which variable is
  independent.
• Ranges from 0 to 1
• Formula
• Lambda E1-E2
• E1

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Instructions to Calculate Lambda

• 1. Calculate E1 Find the mode of the dependent
  variable (the attribute that occurs the most
  often) and subtract it from N (sample size).
  E1N-ƒ of the mode
• 2. Calculate E2 Find the mode in each row (i.e.,
  category of the independent variable). Subtract
  each value from the row (category) total and add
  them together. E2(Row total row mode) (Row
  total row mode) for all attributes of the
  independent variable.

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Are homeless people reporting mental health
problems more likely to request case management
than those who dont?
No Mental Health Mental Health TOTAL
Does Not Want Case Mgt 355 293 648
Wants Case Mgt 157 359 516
Total 512 652 1164
E1 1164-648516 E2 (512-355)
(652-359)450 Lambda.128
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Gender
N
Female 642 73.2
Male 235 26.8
Total 882 100.0
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How likely is it that you will have sexual
intercourse in the next year?
N
1 I definitely will 123 15.3
2 I probably will 149 18.5
3 I don't know 183 22.7
4 I probably will not 87 10.8
5 I definitely will not 263 32.7
Total 805 100.0
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How likely is it that you will have sexual
intercourse in the next year? By Gender

Total
1 female 2 male Total
1 I definitely will 11.5 25.7 15.2
2 I probably will 17.7 21.0 18.6
3 I don't know 20.6 28.6 22.7
4 I probably will not 12.2 7.1 10.8
5 I definitely will not 38.0 17.6 32.7
Total Total 592 210 802
Total Total 100.0 100.0 100.0
?2 49 p lt .001 Lambda .04
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How likely is it that you will have sexual
intercourse in the next year?By Drink Alcohol
Drank alcohol in the last 30 days Drank alcohol in the last 30 days Drank alcohol in the last 30 days Drank alcohol in the last 30 days Drank alcohol in the last 30 days Drank alcohol in the last 30 days
No No Yes Yes Total Total
N N N
I Will 127 22.4 145 57.8 272 30.8
I Don't Know 128 22.6 52 20.7 180 20.7
I Won't 297 52.5 50 19.9 347 39.7
Total 552 100.0 247 100.0 799 100.0

?2 108 p lt .001 Lambda .21
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Testing Hypotheses
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Steps in Conducting Hypothesis Testing

1. State the null hypothesis and alternative.
2. Determine if the test will be one or two tailed
3. Determine the level of measurement of your
   variables
4. Set the alpha level (consider power of the test).
5. Identify the statistical test and assumptions for
   each relationship.

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Common Distributions

• Population Distribution
• Sample Frame
• Sample Distribution
• Sampling Distribution

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Common Sampling Distributions

• Chi Square
• Students t
• F Distribution
• Normal Distribution

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Chi Square

• Chi square is computed based on a comparison of
  actual frequencies observed in a sample to that
  which would be expected to occur by chance alone.
  If there is a large difference between the
  observed vs. the expected frequencies, a large
  value for Chi square will be obtained.

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T-test

• Definition The t-test is used to determine
  whether the difference between means of two
  groups or conditions is due to the independent
  variable, or if the difference is simply due to
  chance.
• (test of independence and paired samples tests)

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One-way ANOVA

• Definition As with the t-test, ANOVA also tests
  for significant differences between groups. But
  while the t-test is limited to the comparison of
  only two groups, one-way ANOVA can be used to
  test differences in three or more groups.

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Sexual Behavior IntentScale Score(5 High Risk)
Drank alcohol in the last 30 days Mean Std. Deviation N
no 2.3862 .95639 563
yes 3.2723 .91401 251
Total 2.6595 1.02802 814
P lt .001 Eta2 .16
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Sexual Behavior IntentScale Score
ANOVA ANOVA ANOVA ANOVA ANOVA ANOVA

Sum of Squares df Mean Square F Sig.
Between Groups 136.301 1 136.301 153.101 .000
Within Groups 722.901 812 .890
Total 859.203 813
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Other Interesting Terms

• Assumption (e.g., normal distribution)
• Assumption Robustness (Leeway one has in
  violating an assumption)

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Type I and Type II Errors

• Type I Rejecting a null hypothesis when it is
  true Saying groups differ when they do not.
• Type II The probability of accepting a null
  hypothesis when it is false Saying groups do
  not differ when they do.
• Power The probability of rejecting a false null
  when it is false the probability of making a
  correct decision.

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Setting Type I and Type II Errors

• H0 Drug is unsafe.
• H1 Drug is safe.
• H0 Defendant is innocent.
• H1 Defendant is guilty.

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Alpha, Beta, and Power(N 15)

• a ß 1-ß
• .10 .37 .63
• .05 .52 .48
• .01 .78 .22

Stevens Applied Multivariate Statistics for the
Social Sciences
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Power and N size

• n (subjects per group) power
•  
• 10 .18
• 20 .33
• 50 .70
• 100 .94

Stevens Applied Multivariate Statistics for the
Social Sciences
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Exercise True or FalseTo achieve the same
power

• More subjects are needed for a 1 level test than
  for a 5 level test.
• Two-tailed tests require larger sample sizes than
  one-tailed tests.
• The smaller the critical effect size, the larger
  the necessary sample size.
• The larger the power required, the larger the
  necessary sample size.
• The smaller the sample size, the smaller the
  power the greater the chance of failure.