MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C8005F.91609C20" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C8005F.91609C20 Content-Location: file:///C:/4919648C/252anoval.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" ANOVA, REGRESSION, CORRELATION

252anova 1/26/07   <= span style=3D'color:red'>(Open this document in 'Outline' view!)                   =     Roger Even Bove

F. ANALYSIS OF VARIANCE  &nbs= p;   

1. 1-Way Analysis of Variance

a. The ANOVA model  -  relation to regression

The one-way ANOVA model is used to compare the means o= f more than two samples, taken from populations that are all assumed to have the same variance. Each sample (ca= lled a treatment) is usually represented as a column= , but there is no requirement that each column have the same number of items = in it.

We will assume the model , where , and <= !--[if gte vml 1]> .We thus have  treatments<= st1:PersonName w:st=3D"on">,  items in ea= ch column and a total of  observation= s. (Thus should be the number in column and row )

b. An ANOVA problem

 

        &= nbsp;       The following data describes monthly expenses for energy in three random sample= s of essentially identical homes. Each column represents expenses on one fuel..

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

Fuel

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

  1

  2

  3

&n= bsp;

&n= bsp;

&n= bsp;

  89

104

  86

&n= bsp;

&n= bsp;

&n= bsp;

101

120

  98

&n= bsp;

&n= bsp;

&n= bsp;

  87

  98

100

&n= bsp;

&n= bsp;

&n= bsp;

  87

110<= o:p>

  96

&n= bsp;

&n= bsp;

Sum

364

432

380

Our hypot= heses are  In the notation used here,   is replaced= by a dot to indicate that a mean has been taken, that is is the mean of column&nb= sp; , in particu= lar, the mean of column 1 is ,  and The overall or grand mean, is the mean of all the numbers in the problem,<= /st1:PersonName> and is often indicated by , but <= !--[if gte vml 1]>  seems to be= a more appealing notation. .

We comput= e three sums of squares.

   (i) The total sum of squares= is the same thing as the numerator of the sample variance of the numbers in the problem.


&nbs= p;

  (ii) The sum of squares within treatments has the same number of terms, but highlights the contribution to the total sum of squares generated by the difference between the individual numbers and the column (treatment) means.=   (iii)= The sum of squares between treatments also has the same number of terms, but it highlights the contribution to the to= tal sum of squares generated by the difference between the column (treatment) m= eans and the overall mean.

 

But, because of the repetition of the column mean= , this can be simplified to .

But note = that , so that the computation of one of the three sums of squares is unnecessary. The materia= l is summarized in a table like the one below.

&n= bsp;

Source

   SS

 DF

  MS

      F

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

Between=

&n= bsp;

   =

&n= bsp;

=

=

=

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

Within<= /p>

&n= bsp;

   =

 

=

=

&n= bsp;

&n= bsp;

&n= bsp;

 

Total

   =

=

 

 

 

 

We fill in the table with the numbers we have computed= and compare the  that we have computed with an  with the appropriate significance level and degrees of freedom shown in the column. If the  that we have computed is larger than the table , reject the= null hypothesis.

&n= bsp;

Source

  SS

 DF

   MS

   &= nbsp; =

        =

        =

&n= bsp;

Between=

   632

   2

   316

   5.51

= s

Column = means equal

&n= bsp;

Within<= /p>

   516

   9

   57.333

&n= bsp;

 

&n= bsp;

&n= bsp;

Total

 1148

 11

&n= bsp;

&n= bsp;

 

&n= bsp;

The ‘s’ for ‘significant difference’ indicates that the null hypothesis of equality of means has been rejected. ‘ns’ for ‘no significant difference’ would indicate that the null hypoth= esis has not been rejected.

&nbs= p;

c. A format for ANOVA

If we use the same simplifications that we use in calculating a sample variance,= we can get the tableau below.

 

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

Fuel

&n= bsp;

  Sum

&n= bsp;

&n= bsp;

&n= bsp;

   1

   2

   3

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

   89

 104

   86

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

 101

 120

   98

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

   87

   98

 100

&n= bsp;

&n= bsp;

&n= bsp;

&n= bsp;

   87

 110

   96

 

&n= bsp;

&n= bsp;

Sum

 364 +=

 432 +=

 380=

=

&n= bsp;

&n= bsp;

=

    4 +

    4 +

    4

=

&n= bsp;

&n= bsp;

= =

 91.00=

108.00<= sub>=

 95.00=

=

&n= bsp;

&n= bsp;

SS

33260= +

46920= +

36216=

=

&n= bsp;

&n= bsp;

=

  8281= +

11664= +

 9025=

=

&nbs= p;

 

Source

  SS

 DF

   MS

     =

        =

        =

&n= bsp;

Between=

   632

   2

   316

   5.51

= s

Column = means equal

&n= bsp;

Within<= /p>

   516

   9

   57.333

&n= bsp;

 

&n= bsp;

&n= bsp;

Total

 1148

 11

&n= bsp;

&n= bsp;

 

&n= bsp;

        &= nbsp;       Explanation: Since the Sum of Squares (SS) column must add up, 516 is found by subtracting  6= 32 from 1148. Since , the total degrees of freedom are . Since there are 3 random samples or columns, the degrees of freedom for Between is 3 R= 11; 1 =3D 2. Since the Degrees of Freedom (DF) column must add up, 9 =3D 11 – 2. The Mean Square (MS) column is found by dividing the SS column by the DF column. 316 is  and 57.333 = is . , and is  compared with  from th= e F table . To see this as Minitab output go to 252anovaex1= .

 

d. Confidence Intervals

i. A single Confidence Interval

If we desire a single interval, we use the formula for the difference between two means when the variance is known.  For example, if we want the difference between means of c= olumn 1 and column 2.  , where .

 

ii. Scheffé Confidence Interval

If we desire intervals that will simultaneously be val= id for a given confidence level for all possible intervals between column means, use .

iii. Bonferroni Confidence Interval

If we only need k different intervals, use

iv. Tukey Confidence Interval

This also applies to all possible differences.

 This gives = rise to Tukey’s HSD (Honestly Significant Difference) procedure. Two sample means  and  are signifi= cantly different if  is greater = than

2. 2 -Way Analysis of  Var= iance

a. The 2-way model

We will assume  rows,  columns and=  observation= s per cell. Thus our model  reads ,  where  .  We will be testing three pairs of Hypotheses -  (i) All row means equal (All zero); Not all row means equal , (ii)  All column = means equal (All zero);   Not a= ll column means equal, (iii)  No interact= ion (All zero) ; Interaction.

This is similar to one-way ANOVA with groups , but= the ‘between’ variation is itemized as to whether it is due to variation between row means, variation between column means or interaction. If we remember that  and , we can rew= rite the one way ANOVA table diagram on the previous table as below. As previous= ly, we get the items in the MS column by dividin= g  the numbers in the SS column, by the numbers in the DF column. The F is th= en found by dividing  by .

&n= bsp;

Source

      SS

      DF

  MS

    = F

=

       &nbs= p;     =

&n= bsp;

Between=

   =

=

=

  ___

___

Treatme= nt means equal

&n= bsp;

Within<= /p>

   =

=

=

&n= bsp;

&n= bsp;

&n= bsp;

 

Total

   =

=

 

 

 

 

We can now rewrite the same table with the ‘between’ items split up.

&n= bsp;

Source

      SS

      DF

  MS

    = F

=

       &nbs= p;     =

&n= bsp;

Rows

   =

   =

=

  ___

___

Row mea= ns equal

&n= bsp;

Columns=

   =

   =

=

  ___

___

Column = means equal

&n= bsp;

Interac= tion

   =

=

=

  ___

___

No Inte= raction

&n= bsp;

Within<= /p>

   =

=

=

&n= bsp;

&n= bsp;

&n= bsp;

 

Total

   =

=

 

 

 

 

 

b. An example

 

 

Insulation 1 (Factor = )

Insulation 2 (Factor = )

 

Fuel 1 (Factor = )

  89

101

  87

  87

 

Fuel 2 (Factor = )

120

110

  98

104

 

Fuel 3 (Factor = )

100

  98

  86

  96

This problem has  rows, columns and, within each cell  measurement= s. We can compute a table of means which shows means for each cell, row and column,<= /st1:PersonName> as well as an overall mean.

 

 

Insulation 1 (Factor = )

Insulation 2 (Factor = )

Row means

 

Fuel 1 (Factor = )

=

=

=

 

Fuel 2 (Factor = )

=

=

=

 

Fuel 3 (Factor = )

=

=

=

 

Column Means

=

=

=

Now we do the computation of sums of squares, using the same simplification that we use in computing a sample variance.

, but we do = not compute this because , so that


Out ANOVA table is thus:

&n= bsp;

Source

    SS

    DF

  MS

    = F

=

       &nbs= p;     =

&n= bsp;

Rows =

  632

     2

  316

  9.88s

=

Row mea= ns equal

&n= bsp;

Columns= =

  300

     1

  300

9.36s

=

Column = means equal

&n= bsp;

Interac= tion =

    24

     2

    12

0.38ns<= /p>

=

No Inte= raction

&n= bsp;

Within<= /p>

   192

     6

    32

&n= bsp;

&n= bsp;

&n= bsp;

 

Total

 1148

   11

 

 

 

 

So we rejectand,but do not = reject.

To explain further, We get the degrees of freedom for rows by taking the number of rows minus 1= ,. We do the same for columns. Then the intera= ction degrees of freedom are the product of row and column degrees of freedom. The total degrees of freedom comes from subtracting 1 from the total number  of items in the problem. The ‘within’ degrees of freedom comes from subtracting the other degrees of freedom from the ‘total’ degrees of freedom.  The ‘MS’ column comes = from dividing the SS column by the DF column. The ‘F’ column is calculated by dividing the items in the ‘MS’ column by .

        &= nbsp;       To see this as Minitab output go to 252anovaex2= . An example of 2-way ANOVA with one measurement per cell is in  252anovaex3= .

&nbs= p;

 

 

c. Confidence Intervals

i. A Single Confidence Interval

If we desire a single interval we use the formula for a Bonferroni Confidence Interval below with .

 

ii. Scheffé Confidence Interval

If we desire intervals that will simultaneously be val= id for a given confidence level for all possible intervals between means, use the following formulas.

 

For cell means, use .        =             &nb= sp;    

        =             &nb= sp;           

For row means, use .

For column means, use

 

Note that if  , replace  with <= !--[if gte vml 1]> .

 

 

iii. Bonferroni Confidence Interval

If we only need m different intervals, use for cell means .

Use for row means .

Use for column means .

iv. Tukey Confidence Interval

For cell means, use .        =             &nb= sp;    

        =             &nb= sp;           

For row means, use .

For column means, use

 

Note that if  , replace  with <= !--[if gte vml 1]> .

 

 

3. More than 2-Way analysis of Variance  S= ee 252anovaex4= .

4. Kruskal-Wallis Test

Equivalent to one-way ANOVA when the underlying distri= bution is non-normal.

Columns come from same distribution or medians equal.

Example: Use same example as for one-way ANOVA<= st1:PersonName w:st=3D"on">, but assume that data comes from non-normal s= ource. Assume that . There are  data items<= st1:PersonName w:st=3D"on">, so rank them from 1 to 12. Let  be the numb= er of items in column and be the rank sum of column . .

 

       &nbs= p;             Original Data

 

       &nbs= p;            &= nbsp; Ranked Data

 

Treatment

      1

Treatment

       2

Treatment

      3

 

Treatment

      1

Treatment

       2

Treatment

      3

 

   &= nbsp; 89

   &= nbsp; 104

   &= nbsp; 86

 

      4

   &= nbsp; 10

      1

 

   101

   &= nbsp; 120

   &= nbsp; 98

 

      9

   &= nbsp; 12

      6.5

 

   &= nbsp; 87

       98=

   100

 

      2.5

       6.= 5

      8

 

   &= nbsp; 87

   &= nbsp; 110

   &= nbsp; 96

 

    =   2.5

   &= nbsp; 11  .

    =   5  .

 

 

 

 

       &nbs= p;  =

    18.0

   &= nbsp; 39.5

    20.5

 

 

 

 

       &nbs= p;    =

      4

       4<= /p>

       4<= /p>

To check the ranking, note that the sum of the three rank sums is 18.0 + 39.5 + 20.5  =3D 78.0, and  that the sum of the first=  numbers is =  

 

Now, compute the Kruskal-Wallis statistic  . If we look up this result in the (4, 4, 4) section of the Kruskal-W= allis table (Table 9) , we find that= the p-value for is .054 and that the p-value for is .097, so = the p-value for must lie between these two. Since both are above , do not rej= ect .

If the size of the problem is larger than those shown = in Table 9, use the distribution, with , where is the number of columns. For example, if each of  columns con= tains 6 items, and ,  compare with . Since is smaller than , do not rej= ect the null hypothesis.

 

5. Friedman Test

Equivalent to two-way ANOVA with one observation per c= ell when the underlying distribution is non-normal.

 Columns com= e from same distribution or medians equal. Note that the only difference between t= his and the Kruskal-Wallis test is that the data is cross-classified in the Friedman test.

Example: Three groups of 4 matched workers are trained to do a task by four different methods. When each worker is observed later, he or she is given a gr= ade of 1 through 10 on performance of the task. Note that because this data is ord= inal, ANOVA is not appropriate. Assume that <= !--[if gte vml 1]> . In the data below, the methods are represented  b= y columns, and= the groups by rows.. In each row the numbers are ranked from 1 to = . For each column, compute, the rank s= um of column .

 

 

 

       &nbs= p;            &= nbsp;   Original Data

 

       &nbs= p;            &= nbsp;   Ranked Data

 

 

Method

      1

Method

      2

Method

      3

Method

      4

 

Method

      1

Method

      2

Method

      3

Method

      4

 

Group 1

      9

      4

      1

      7

 

      4

      2

      1

      3

 

Group 2

   &= nbsp;  6

      5

      2

      8

 

      3

      2

      1

      4

 

Group 3

      9

      1

      2

      6

 

    =   4

    =   1

    =   2

    =   3

 

 

 

 

 

 

=

    11

      5

      4

    10

To check the ranking, note that the sum of the four rank sums is 11 + 5 + 4 + 10 =3D 30, and  that the sum of the   numbers in = a row is . However, t= here are  rows, so we must multiply the expression by <= !--[if gte vml 1]> . So we have .

 

Now compute the Friedman statistic   = .  If we fin= d the place on the Friedman Table (Table 8) for 4 columns and 3 rows, we find that the p-value for  is .033. Si= nce the p-value is below , reject the= null hypothesis.

If the size of the problem is larger than those shown = in Table 10, use the distribution, with , where= is the number of columns. For example, if each of  columns con= tains 6 items, and , compare  with <= !--[if gte vml 1]> . Since is not larger than , do not rej= ect the null hypothesis.

 

6. Tests for Equality of Variances – See 252mvar.

 

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