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