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Statistical Methods

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Statistical Methods Variability and Averages The Normal Distribution Comparing Population Variances Experimental Error & Treatment Effects Evaluating the Null Hypothesis – PowerPoint PPT presentation

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Title: Statistical Methods


1
Statistical Methods
  • Variability and Averages
  • The Normal Distribution
  • Comparing Population Variances
  • Experimental Error Treatment Effects
  • Evaluating the Null Hypothesis
  • Assumptions Underlying Analysis of Variance

2
Variability and Averages
Patients
Controls
  • Graph 1 Bipolar disorder
  • Different variability
  • Same averages
  • Graph 2 Blood sugar levels
  • Same variability
  • Different averages

Frequency
Depressed Manic
Patients
Controls
Frequency
Low
High
3
The Normal Distribution
  • The normal distribution is used in statistical
    analysis in order to make standardized
    comparisons across different populations
    (treatments).
  • The kinds of parametric statistical techniques we
    use assume that a population is normally
    distributed.
  • This allows us to compare directly between two
    populations

4
The Normal Distribution
  • The Normal Distribution is a mathematical
    function that defines the distribution of scores
    in population with respect to two population
    parameters.
  • The first parameter is the Greek letter (m, mu).
    This represents the population mean.
  • The second parameter is the Greek letter (s,
    sigma) that represents the population standard
    deviation.
  • Different normal distributions are generated
    whenever the population mean or the population
    standard deviation are different

5
The Normal Distribution
  • Normal distributions with different population
    variances and the same population mean

6
The Normal Distribution
  • Normal distributions with different population
    means and the same population variance

f(x)
7
The Normal Distribution
  • Normal distributions with different population
    variances and different population means

2
s
1
1
m

2
s
3
3
m

8
Normal Distribution
  • Most samples of data are normally distributed
    (but not all)

9
Comparing Populations in terms of Shared Variances
  • When the null hypothesis (Ho) is approximately
    true we have the following
  • There is almost a complete overlap between the
    two distributions of scores

10
Comparing Populations in terms of Shared Variances
  • When the alternative hypothesis (H1) is true we
    have the following
  • There is very little overlap between the two
    distributions

11
Shared Variance and the Null Hypothesis
  • The crux of the problem of rejecting the null
    hypothesis is the fact that we can always
    attribute some portion of the difference we
    observe among treatment parameters to chance
    factors
  • These chance factors are known as experimental
    error

12
Experimental Error
  • All uncontrolled sources of variability in an
    experiment are considered potential contributors
    to experimental error.
  • There are two basic kinds of experimental error
  • individual differences error
  • measurement error.

13
Estimates of Experimental Error
  • In a real experiment both sources of experimental
    error will influence and contribute to the scores
    of each subject.
  • The variability of subjects treated alike, i.e.
    within the same treatment condition or level,
    provides a measure of the experimental error.
  • At the same time the variability of subjects
    within each of the other treatment levels also
    offers estimates of experimental error

14
Estimate of Treatment Effects
  • The means of the different groups in the
    experiment should reflect the differences in the
    population means, if there are any.
  • The treatments are viewed as a systematic source
    of variability in contrast to the unsystematic
    source of variability the experimental error.
  • This systematic source of variability is known as
    the treatment effect.

15
An Example
  • Two lecturers teach the same course.
  • Ho lecturer does not influence exam score.
  • Experimental design
  • 10 students 5 assigned to each lecturer.
  • IV Lecturer (A1, A2)
  • DV Exam score
  • Results
  • A1 16, 18, 10,12,19
  • A1 Mean15
  • A2 4, 6, 8, 10, 2
  • A2 Mean6

16
Partitioning the Deviations
17
Partitioning the Deviations
  • Each of the deviations from the grand mean have
    specific names
  • is called the total deviation.
  • is called the between groups
    deviation.
  • is called the within subjects
    deviation.
  • Dividing the deviation from the grand mean is
    known as partitioning

18
Evaluating the Null Hypothesis
  • The between groups deviation
  • represents the effects of both error and the
    treatment
  • The within subjects deviation
  • represents the effect of error alone

19
Evaluating the Null Hypothesis
  • If we consider the ratio of the between groups
    variability and the within groups variability
  • Then we have

20
Evaluating the Null Hypothesis
  • If the null hypothesis is true then the treatment
    effect is equal to zero
  • If the null hypothesis is false then the
    treatment effect is greater than zero

21
Evaluating the Null Hypothesis
  • The ratio
  • is compared to the F-distribution

22
ANOVA
  • Analysis of variance uses the ratio of two
    sources of variability to test the null
    hypothesis
  • Between group variability estimates both
    experimental error and treatment effects
  • Within subjects variability estimates
    experimental error
  • The assumptions that underly this technique
    directly follow on from the F-ratio.

23
Assumptions Underlying Analysis of Variance
  • The measure taken is on an interval or ratio
    scale.
  • The populations are normally distributed
  • The variances of the compared populations are the
    same.
  • The estimates of the population variance are
    independent
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