# Statistics ______

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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 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
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
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

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

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
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
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
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

-

II I
qduifg
0
3
3
70 a

I
0 n
1
0
00
1
3
7
0
3
1
3
7
0
3
1
3
7
1
3
8
0
0
2
0
1
0
a

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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