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					          PSYM021
Introduction to Methods & Statistics


Week Three: Statistical techniques I

            Cris Burgess
                 Reporting statistics

   It isn’t enough to insert a table and leave it at that
   You must follow APA guidelines:
        Use tables, but also describe what they contain
        Draws the reader’s attention to important features
   ‘Levels of reading’ journal articles and lab reports
                Types of statistics
   Descriptive statistics
    Summarise the data (‘shorthand’)
        ‘Describe the symptoms’

   Inferential statistics
    Look at relationships within the data
    Draw ‘inferences’
        ‘Establish the possible causes’
        Indicate how ‘significant’ the results are
                 Inferential statistics

   What’s the answer?
   What’s the question?
   May sound like stating the bleeding obvious, but…
   Establishing the specific research question to test is
    the hardest thing in research
   What is the relationship between A and B?
        How are A and B measured?
        How can differences/similarities be tested?
               Inferential Statistics
   Logic behind the test
   Concept, not calculation
   Some commonly-encountered tests
           Tests of Frequency
           Tests of Differences
           Tests of Association
                 Inferential statistics
   How can we compare variable scores against one another?
   Frequency - compares frequency of membership of one
    category with another (nominal or ordinal)
     e.g. Chi-square test
   Differences - compares two groups in terms of a ‘score’
    (interval or ratio)
     e.g. T-test
   Association - compares two variables in terms of their
    ‘degree of association’ (ordinal, interval or ratio)
     e.g. Correlation
        Test of frequency - Chi-square,                  χ 2

   Compare two category variables for degree of association (non-
    parametric)
       eg: gender (male/female) vs behaviour (yes/no)

                             Behaviour:
                             No     Yes       Total:
         Gender:    Male      60     120       180
                   Female     70      20        90
                   Total:    130     140       270
                       χ2 = 6.5, p=0.011
        Test of frequency - Chi-square, χ2
   Test statistic is “χ2”
   Hypothesis is “Men do it more than women”
   Test might be ‘significant’, but what does that mean?
   Compare observed frequencies with those expected by chance

                            Behaviour:
                           No       Yes      Total:
      Gender:   Male     60Low      High
                             (90) 120 (90)    180
                           (45) 20Low
                Female 70High       (45)       90
                Total:    130        140      270
                    χ2 = 6.5, p=0.011
        Test of frequency - Chi-square, χ2
   More categories for ‘Yes’ response (ordinal variable)
   Hypothesis is “Men do it more than women”
   Now test might not be ‘significant’, but what does that mean?
   Compare observed frequencies with those expected by chance

                                Behaviour:
                          No      A little   A lot   Total:
      Gender:   Male        (87) High   High
                         60Low 60 (47) 60 (47)        180
                         High   Low     Low
                Female 70 (43) 10 (23) 10 (23)        90
                Total:    130       70        70      270

                          χ2 = 4.74, p=0.093
             Frequency distribution

                               9
   Normally distributed       8
                               7
   The area under the line    6

    represents the number of   5
                               4
    cases (respondents,        3
    participants) recording    2
    each score on the DV       1
                               0
                                   1   2   3   4   5   6   7
         Test of difference – T-test
   T-test
      Compares two groups
      Interval or ratio dependent variable

      Assumes ‘parametric’ frequency distribution

   “Effect of alcohol on reaction time”
      ‘Alcohol’ vs ‘No alcohol’ groups (IV)
      Reaction time (DV) is an interval variable

      Reaction time (DV) is normally distributed

   Meets the requirements of the test
   Hypothesis - “Alcohol will affect Reaction time”
   OR: “Reaction times will be slowed by alcohol”
    Variance ratio: the “t” statistic
Broadly speaking:
                             Area of non-overlap
     “t” statistic =
                               Area of overlap

                         Area where hypothesis true
     “t” statistic =
                         Area where hypothesis false


      Large “t” suggests statistically ‘significant’
             difference between groups

        This is often called the ‘variance ratio’
            Hypothesis test: ‘Alcohol’ vs ‘No alcohol’ condition

                Hypothesis true (reaction time slower in ‘alcohol’ condition)
                Hypothesis false (reaction time faster in ‘alcohol’ condition)


                                  Effect of alcohol on RT
Frequency




                   No alcohol                                 Alcohol




                                                            Reaction time (ms)
                       Test of difference – T-test
           Test statistic is “t”
                               non-overlap (hypothesis true)
                                  overlap (hypothesis false)

                                            Effect
Frequency




                    No alcohol                                 Alcohol




                                                         Reaction time (ms)
                       Test of difference – T-test
           Test statistic is “t”
                               non-overlap (hypothesis true)
                                  overlap (hypothesis false)

                                    Effect
Frequency




                    No alcohol                Alcohol

                                                        Smaller difference
                                                         between mean RT for
                                                         each condition



                                                         Reaction time (ms)
                              Variance
   Referred to earlier as                            Mean R.T. (ms)




                                     Frequency
    measure of ‘dispersion’
   Variance describes the
    variability in scores within a
    group
                                                 Reaction time (ms)
   Top diagram shows large
    variance (reflected in large

                                     Frequency
    standard deviation)
   Bottom diagram shows small
    variance (reflected in small
    standard deviation)
                                                 Reaction time (ms)
                              Large dispersion
           Test statistic is “t”
                               non-overlap (hypothesis true)
                                  overlap (hypothesis false)

                                    Effect
Frequency




                    No alcohol                Alcohol
                                                           Large standard
                                                            deviation/range for
                                                            groups’ reaction
                                                            times



                                                            Reaction time (ms)
                             Smaller dispersion
           Test statistic is “t”
                               non-overlap (hypothesis true)
                                  overlap (hypothesis false)

                                    Effect
Frequency




                    No alcohol                Alcohol

                                                     Smaller standard
                                                      deviation/range for
                                                      groups’ reaction times



                                                         Reaction time (ms)
                              Design
   Measures may be:
     Repeated (same participants in each condition)

     Non-repeated (different participants in each condition)


   Why is this important?
   Same participants (repeated measures)
      can directly compare Joe Blogg’s score in each condition

      can assume same shape of distribution in each condition

      in other words, same dispersion, different mean

   Different participants (non-repeated measures)
      must summarise differences between conditions

      cannot assume the same shape of distribution in each condition

      in other words, different dispersion in each condition
                                        Design
   Same participants (repeated measures)
                                         Effect
      Frequency


                      Before alcohol              After alcohol




                                                           Reaction time (ms)
   Different participants (non-repeated measures)
                                         Effect
          Frequency




                       Before alcohol               After alcohol




                                                            Reaction time (ms)
                  Analysis of Variance (ANOVA)
           Test statistic is “F”
                               non-overlap (hypothesis true)
                                 overlap (hypothesis false)
Frequency




                                      1 unit 2 units   5 units     10 units
                     No alcohol




                                                           Reaction time (ms)
     Variance ratio: the “F” statistic

               Area of non-overlap (hypothesis true)
“F-ratio” =
                 Area of overlap (hypothesis false)

  • Large “F” means significant differences
  • Large “F” means evidence in support of hypothesis
  • Need to calculate size of all these ‘areas’
           ANOVA - important equations
                 SS
  MS =                              Where:
                 df                 MS - ‘mean square’

                MSvariable          SS - ‘sum of squares’
      F =                           df - ‘degrees of freedom’
                MSerror
• MSvariable - the variance accounted for by the variable
• MSerror - the variance not accounted for by the variable
• F ratio is a variance ratio or ‘signal to noise’ ratio
• Large F means large differences accounted for by the variable
• Statistical significance of F ratio is reported by SPSS
                     No alcohol     1 unit   2 units   5 units
Frequency




                                                                  Reaction time (ms)

           ANOVA will tell us if one condition is significantly different to one
            or more of the others
           But it won’t tell us which conditions are different
           We can compare one (or more) against one (or more) of the others
           Need to run “contrasts”, using “contrast coefficients”
                         Bags of flour
    Bag1         Bag2         Bag3         Bag4         Bag5




Five bags of flour - how heavy ?
Using contrast coefficients, we can compare one against another,
while ignoring the rest
       Which is Heavier - Bag 2 or Bag 5?
Bag1             Bag2        Bag3          Bag4      Bag5




                   -1                      +1
                               0




          Contrast Weights are: (0, Bag 4 Bag
       Position: Bag1 Bag 2 Bag 3 -1, 0, 0, +1) 5
       weight:     0    -1         0   0        +1
Does the average weight of 1 & 2 differ from 3 ?
Bag1         Bag2         Bag3        Bag4          Bag5




                                   Place Bag 3 at twice
       Answer….
       What weights would you use?
       Why?                        the distance (has same
       Contrast weights: (?, -1, 2, 0, as 4 & 5
                                    ?, Ignore
       Contrast weights: (-1, ?, ?,effect 0) placing -two
                                        ?)
                                       by 3s on normal
                                      Bag placing them
                                       scales…)
                                       at zero (pivot)

                -2   -1     0    +1   +2     So: (-1,-1, 0, 0)
                                             So: (?, ?, ?,2, 0, 0)
                                              so far….
Does the average weight of 1, 3 & 5 together
     differ from the average of 2 & 4 ?

Bag 1     Bag 2     Bag 3       Bag 4     Bag 5




   Answer….
   What weights would you use?
                     (?, +3, ?, +3,
   Contrast weights: (-2,?, ?, -2,?) -2)
   Why?
                                      “Polynomial” contrasts

                                                 Does recall fall with delay?
                     15
                                                 What contrast weights should
Recall performance




                     10                          be used to answer this question?

                                                 Suggestions?….
                      5

                                                   Answer: (-3, -1, 1, 3)
                      0
                           1      2      3   4     …which are the weights for
                                                   a linear polynomial contrast
                          Increasing Delay
                Polynomial contrasts
   The significance of the linear contrast tells us whether the
    best-fitting straight line slopes downwards (or upwards!)
   A quadratic contrast (showing whether the best-fitting
    quadratic curve has a “bend” in it) would indicate whether
    the data plot departs from linearity (Weights for 4 levels:
    (1, -1, -1, 1)
   A cubic contrast picks up evidence for an S-shaped bend
    (double inflection) (Weights for 4 levels: (-1, 3, -3, 1)
   Weights to be used in different designs can be found on the
    back page of the stats booklet.
                          Does recall fall with delay, but then recover?

                     18
                     16                                   What contrast weights should
                     14                                   be used to answer this question?
Recall performance




                     12
                     10                                   Suggestions?….
                     8
                     6
                     4
                                                         Answer: (1,-1,-1, 1)
                     2
                     0                                   …which are the weights for
                          0      1     2    3    4   5   a quadratic polynomial contrast

                              Increasing Delay
                          Does recall fall with delay, but only once a
                                  critical point is reached?
                     16
                     14                                  What contrast weights should
                     12                                  be used to answer this question?
Recall performance




                     10
                     8                                   Suggestions?….
                     6
                                                         Answer: (-1, 3, -3, 1)
                     4
                     2
                                                         …which are the weights for
                     0
                                                         a cubic polynomial contrast
                          0        2      4      6   8


                              Increasing Delay
                                 Polynomial relationships
 14                                                       16
                                                          14
 12
 10                                            Linear     12                                         Cubic
                                                          10
     8
                                                          8
     6
                                                          6
     4
                                                          4
     2                                                    2
     0                                                    0
         0       2       4        6       8                    0       2       4       6        8



18                                                        16
16                                                        14
14                                            Quadratic   12
                                                                                                    Quartic
12
                                                          10
10
8                                                         8
6                                                         6
4                                                         4
2
                                                          2
0
     0       1       2       3    4   5                   0
                                                               0   2       4       6       8   10
                              Break
   Phew! I need a drink...
        One-way ANOVA: Driving study
• Imagine an experiment on people’s estimations of vehicle
  distances when driving under four different circumstances:
   (1) Day-time driving under Clear weather conditions
   (2) Night driving under Clear weather conditions
   (3) Day-time driving under Foggy weather conditions
   (4) Night driving under Clear weather conditions
• Suppose three drivers were tested on motorways in each case
• Independent variable = driving condition
 Four categories or “levels”
• Dependent variable = accuracy of distance estimate
  1 = random guessing          10 = perfect performance
        One-way ANOVA: Driving study
DV = “Distance estimate accuracy” IV = “Driving condition”
 RESULTS:
 Day/Clear     Night/Clear   Day/Foggy                      Night/Foggy
    9               8             5                                2
   10               9             6                                1
    9               7             5                                1
   28             24             16                                4 (Totals)
                                         30      28

                                         25                 24


 Is there a statistically                20
                              Accuracy

                                                                       16
 significant difference                  15

 between the conditions?                 10

                                         5                                         4

                                         0
                                              Day-clear Night-clear Day-foggy Night-foggy
            One-way ANOVA: Driving study
                                                 Strong effect of
     ANOVA over the four groups gives:
                                                Driving condition
                                                -
                                                       on
                                                Estimate accuracy
I          I I
           if
                           a

        I




    • But this just means that performance in one or more of the
      groups was different from that in the others.
    • It does NOT mean that performance is affected by Day versus
      Night driving, or by driving in Clear versus Foggy conditions.
                        Two-way ANOVA: Driving study
           30      28                                                         30
                                                                                                  26
           25                 24
                                                                              25   22




                                                              mean accuracy
           20                                                                 20
Accuracy




                                         16
           15                                                                 15          14
                                                                                                          10
           10                                                                 10

           5                                         4
                                                                              5

           0                                                                  0
                Day-clear Night-clear Day-foggy Night-foggy                        Day   Night   Clear   Foggy

• Average performance in Day driving (mean = 22) is better than
  that for Night driving (mean = 14)
• Average performance in Clear driving (mean = 26) is better than
  that for Foggy driving (mean = 10)
• The drop in performance produced by fogginess is greater at
  Night (20-point drop) than it is for Day driving (12-point drop).
  How can we provide statistical support for these observations?
  Results of Two-way ANOVA on original data

                                     -


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                                     The contrast
                The usual suspects   F-ratios
                     Implications

   This means that you can use ANOVA to examine the
    independent effects on your data of 3, 4 or more
    different independent variables
   In the SPSS output from such analyses, the so-called
    main effects tell you whether each variable has an
    independent effect (by itself) on the dependent variable
   The interactions tell you how the effect of each
    independent variable is itself altered by the influence of
    the others
                                              Examples: Interaction effects
                                30                                           40
Accuracy of distance estimate




                                        Day
                                                                             35         Extraverts
                                25      Night
                                                                             30                           Introverts




                                                               Performance
                                20
                                                                             25
                                15                     Day                   20                           Extraverts
                                                                             15         Introverts
                                10
                                                                             10
                                5                      Night                 5
                                0                                            0
                                     Clear          Foggy                           Morning          Afternoon


                   • Interaction between weather                             • Interaction between personality
                     conditions and time of day                                type and time of day
                   • The effect of fog on distance                           • Extraverts bouncy in morning,
                     judgements is greater at night                            slow down during course of day
                     than during the day                                     • Introverts take a while to get
                                                                               going
                           Summary
   Non-parametric tests are limited in their ability to provide the
    experimenter with grounds for drawing conclusions - parametric
    tests provide more detailed information
   ‘Tests of difference’ use a statistic that reflects a ‘signal to noise’
    ratio, or how much variance in the DV is accounted for by the
    IV, compared with the what is left
   The only fundamental difference between a t-test and ANOVA
    is the number of levels in the Independent Variable (IV)
   T-tests: IV has two levels; ANOVA: IV has three or more levels
    (or two or more IVs with 2+ levels)
   We can combine a number of IVs together in the same ANOVA
    procedure (two-way, three-way etc.), identifying their individual
    and combined (interaction) effects on the DV

				
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