Psych 218 Introduction to Behavioral Research Methods

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Psych 218Introduction to Behavioral Research
Methods

• Week 13 Lecture 25

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Outline of Todays Lecture

• Last lecture we discussed
• Lab reports and proposal presentation
• Today we will discuss
• Inferential Statistics
• t-tests
• ANOVA
• Correlation and regression

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How is p calculated? It depends on

• the scaling properties of your dependent variable
  (DV)
• DV is interval or ratio? parametric tests
• DV is nominal or ordinal? non-parametric tests
• Research design
• Experimental test differences on measure
  between conditions or groups ? t-test, ANOVA,
  sign test, Mann-Whitney
• Correlational test relations between different
  measures ? Pearson product-moment correlation,
  point-biserial correlation, etc.
• the manner in which you phrase your hypotheses
• One tailed vs. two-tailed tests

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Statistical Tests and Ratios of Variability

• Conceptually, all statistical tests (t-test,
  ANOVA, Pearson r, etc.) partition variability
  into 2 categories
• Between-treatments (groups) variability
• Within-treatments (groups) variability (error)
• The statistic equals the ratio of these
  categories
• between-treatments variability
• Statistic --------------------------------------
  - ,
• within-treatments variability

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The Standard Error of the Mean

• An estimate of the amount of variability expected
  in the sample means across a series of samples
• For greater sample sizes (N) ? expect less
  variability
• Example IQ test (normally distributed, m 100,
  s 10)
• Random generation of 5 samples of 10 numbers from
  such a normal distribution yields sample means
  
• 100.5, 105.3, 98.76, 98.97, 102.6
• 5 samples of 100 numbers yields sample means
  
• 99.31, 98.26, 100.5, 101.3, 99.44

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The Standard Error of the Mean
Distribution of sample means for small N
Distribution of sample means for large N
Population Distribution (m, s)
Frequency
Frequency
m
m
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The Critical Question

• How do we know whether differences between sample
  means actually arise from the fact that the
  samples are drawn from two different populations
  (created by an effective treatment) or whether
  the differences are simply due to just random
  variability (error) in our measure?
• We cannot know the answer to this question,
  however, using the laws of probability we can
  estimate the probability that the differences
  between the sample means are due to chance

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Example The t-test, the Simplest Case of
Calculating p
Population Distribution (m, s)

• Null Hypothesis this class has average
  intelligence (average class IQ 100)
• Alternative Hypothesis this class does not have
  average intelligence (average class IQ is either
  higher or lower than 100)
• Assume we measure each individuals IQ and
  calculate the following 105.3, s
  8.524, N 50

Frequency
m
Distribution of sample means for N 50
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Example The t-test, the Simplest Case of
Calculating p

• Null Hypothesis this class has average
  intelligence (average class IQ 100)
• Alternative Hypothesis this class does not have
  average intelligence (average class IQ is either
  higher or lower than 100)
• Assume we measure each individuals IQ and
  calculate the following 105.3, s
  8.524, N 50

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So, We have Calculated that t 4.4, What Now?

• Determine the degrees of freedom (df) used to
  calculate t for a single sample df N1 501
  49
• Look up the probability of that value of t
  occurring in a table of the t-distribution based
  on the df (Table 2 of the texts appendix on page
  A-7)
• Because df 49 absent, use row for df 40
• Scan row for last t value lt your calculated value
• Scan up column for the 2-tailed alpha level
• OR, use a computer program like SPSS to estimate p

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One-Tailed or Two-tailed?
Population Distribution (m, s)

• Depends on how the alternative hypothesis is
  phrased
• Alternative Hypothesis this class does not have
  average intelligence (average class IQ is either
  higher or lower than 100) ? two-tailed
• Alternative Hypothesis this class has
  above-average intelligence (average class IQ is
  higher than 100) ? one-tailed

Frequency
m
Distribution of sample means for N 50
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The t-test testing for differences

• t-test can also be used to test for differences
  between 2 groups
• Or with matched-samples (repeated measures)

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Inferential Statistics for Correlation and
Regression

• For studies examining the relationship between
  two dependent variables (DVs), inferential
  statistics assess the reliability of the
  correlation coefficient (r)
• Similar to t-test, computer programs such as SPSS
  determine a p value based on the calculated value
  of r with df N 1

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Inferential Statistics for Correlation and
Regression
Correlation Matrix for 4 DVs (A, B, C, D)

• Correlation Matrix for studies examining more
  than 2 DVs, multiple correlations (rs) result,
  each with a p value
• Number of rs square of number of DVs
• rAA, rBB, rCC, rDD, must always 1.0
• rAB rBA, rAC rCA, etc.

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Inferential Statistics for Correlation and
Regression

• 4 DVs (A, B, C, and D) , six rs rAB, rAC, rAD,
  rBC, rBD, rCD

• Only interested rs below the diagonal
• 3 DVs (A, B, and C), three rs rAB, rAC, rBC

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Beyond Two Groups

• t-tests are only used when comparing 2 groups or
  treatments, when comparing more than 2 groups use
  analysis of variance (ANOVA)
• Just like the t-test, ANOVA partitions all the
  variability in the data into 2 categories
  between-treatment and within-treatment
• The F value equals the ratio of the two
  categories
• between-treatments variability
• F ---------------------------------------
  within-treatments variability

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Interpreting the ANOVA F statistic

• Think of F as equivalent to t, it is a statistic
  with a particular distribution that allows us to
  estimate a value for p, the probability of a Type
  I error
• The p value corresponding to a particular value
  of F depends on 2 parameters representing degrees
  of freedom (df), look it up in a table of F(df1,
  df2)
• df1 is based on the number of treatments
• df2 is based on the number of treatments and the
  number of subjects used

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Single-Factor ANOVA

• Use if study has 1 independent variable with more
  than 2 conditions (levels or a)
• df1 a 1
• df2 a(s 1), s number of subjects in each
  group
• If p lt .05, then 2 or more of the treatment means
  are reliably different but ANOVA doesnt tell you
  which means
• Use planned or unplanned (post-hoc) comparisons
  to determine exactly which treatment means are
  reliably different

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Multi-Factor ANOVA

• Use if study has 2 or more independent variables
  (IVs or factors) with any number of levels
• Multi-factor ANOVA returns multiple F and p
  values, 1 set for each
• IVs main effect the individual effect of a
  given IV that is independent of the effects of
  the other IVs
• Interaction between IVs changes in the effect of
  one IV at different levels of another IV

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Output of Multi-Factor ANOVA

• 2-factor ANOVA returns 3 sets of F and p values
• Two for main effects Factor (IV) A and Factor B
• One two-way interaction A x B
• 3-factor ANOVA returns 7 sets of F and p values
• Three main effects 1 each for Factors A, B, and
  C
• Three 2-way interactions A x B, A x C, and B x
  C
• One 3-way interaction A x B x C

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Example of 2-factor ANOVA

• Hypothesis Using a cell-phone while driving will
  increase breaking response time to a child
  entering the street when driving in busy traffic,
  but not when driving alone.
• Independent Variables (factors)
• Cell Phone Status, 2 levels in use, not in use
• Traffic Status, 2 levels busy traffic, no
  traffic
• Dependent Variable breaking response time

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Example of 2-way ANOVA

• 2-Factor Univariate Analysis of Variance is used
  to analyze the data
• Table of Means and Standard Errors from SPSS

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Example of 2-way ANOVA
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Example of 2-way ANOVA
ANOVA Table from SPSS