PowerPoint Presentation - Overview of Lecture by sdfwerte

VIEWS: 15 PAGES: 23

									Overview of Lecture


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




C82MST Statistical Methods 2 - Lecture 2           1
Variability and Averages
                                                                   Patients
                                                                   Controls
• Graph 1: Bipolar disorder




                                           Frequency
   • Different variability
   • Same averages

                                                       Depressed     Manic

                                                                   Patients
• Graph 2: Blood sugar                                             Controls
                                           Frequency
  levels
    • Same variability
    • Different averages

                                                       Low            High
C82MST Statistical Methods 2 - Lecture 2                                      2
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




C82MST Statistical Methods 2 - Lecture 2                       3
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



C82MST Statistical Methods 2 - Lecture 2                          4
The Normal Distribution


• Normal distributions with different population variances and
  the same population mean


                                           s2 = 1



                                           s2 = 2

                            f(x)           s2 = 3
                                           s2 = 4




C82MST Statistical Methods 2 - Lecture 2                         5
The Normal Distribution


• Normal distributions with different population means and
  the same population variance



                                           m= 1   m= 2




                         f(x)




C82MST Statistical Methods 2 - Lecture 2                     6
The Normal Distribution


• Normal distributions with different population variances and
  different population means



                                           s 2 = 1 m= 1




                                                   s 2 = 3 m= 3




C82MST Statistical Methods 2 - Lecture 2                          7
Normal Distribution


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




C82MST Statistical Methods 2 - Lecture 2                        8
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


C82MST Statistical Methods 2 - Lecture 2                   9
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



C82MST Statistical Methods 2 - Lecture 2                       10
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




C82MST Statistical Methods 2 - Lecture 2                        11
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.




C82MST Statistical Methods 2 - Lecture 2                         12
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




C82MST Statistical Methods 2 - Lecture 2                            13
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.




C82MST Statistical Methods 2 - Lecture 2                          14
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: Mean=15
   • A2: 4, 6, 8, 10, 2
   • A2: Mean=6



C82MST Statistical Methods 2 - Lecture 2           15
Partitioning the Deviations


                           AS 25                 A2                   T


Within Subjects                                                           Between Subjects
deviation                                                                 deviation



                    0      2       4       6         8        10


                                   AS 25 - A 2               A2 - T



                                           AS 25 -       T


                                          Total
                                          Deviation


C82MST Statistical Methods 2 - Lecture 2                                                     16
Partitioning the Deviations


• Each of the deviations from the grand mean have specific
  names
   • AS 25  T is called the total deviation.
   • A2  T is called the between groups deviation.
   • AS 25  A is called the within subjects deviation.
• Dividing the deviation from the grand mean is known as
  partitioning




C82MST Statistical Methods 2 - Lecture 2                     17
Evaluating the Null Hypothesis


• The between groups deviation
           A2  T

• represents the effects of both error and the treatment
• The within subjects deviation
          AS 25  A

• represents the effect of error alone




C82MST Statistical Methods 2 - Lecture 2                   18
Evaluating the Null Hypothesis


• If we consider the ratio of the between groups variability
  and the within groups variability
                           Differences among treatment means
                          Difference among subjects treated alike


• Then we have

                          Experimental Error + Treatment Effects
                                    Experimental Error




C82MST Statistical Methods 2 - Lecture 2                            19
Evaluating the Null Hypothesis


• If the null hypothesis is true then the treatment effect is
  equal to zero:

                                   Experimental Error + 0
                                                          =1
                                     Experimental Error


• If the null hypothesis is false then the treatment effect is
  greater than zero:
                          Experimental Error + Treatment Effect
                                                                1
                                   Experimental Error



               
C82MST Statistical Methods 2 - Lecture 2                             20
Evaluating the Null Hypothesis


• The ratio
                          Experimental Error + Treatment Effect
                                                                1
                                   Experimental Error

• is compared to the F-distribution
              




C82MST Statistical Methods 2 - Lecture 2                             21
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.




C82MST Statistical Methods 2 - Lecture 2                        22
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




C82MST Statistical Methods 2 - Lecture 2                       23

								
To top