analysis of variance (anova) and multivariate analysis of variance (manova)
DESCRIPTION
Analysis of Variance (ANOVA) and Multivariate Analysis of Variance (MANOVA). Session 6. Analysis of Variance. Using Statistics. The Hypothesis Test of Analysis of Variance. The Theory and Computations of ANOVA. The ANOVA Table and Examples. Further Analysis. Models, Factors, and Designs. - PowerPoint PPT PresentationTRANSCRIPT
Analysis of Variance (ANOVA) and Multivariate Analysis of
Variance (MANOVA)
Session 6
• Using Statistics.
• The Hypothesis Test of Analysis of Variance.
• The Theory and Computations of ANOVA.
• The ANOVA Table and Examples.
• Further Analysis.
• Models, Factors, and Designs.
• Two-Way Analysis of Variance.
• Blocking Designs.
• Using the Computer.
• Summary and Review of Terms.
Analysis of Variance
• ANOVA (ANalysis Of VAriance) is a statistical method for determining the existence of differences among several population means.
ANOVA is designed to detect differences among means from populations subject to different treatments.
ANOVA is a joint test
The equality of several population means is tested simultaneously or jointly.
ANOVA tests for the equality of several population means by looking at two estimators of the population variance (hence, analysis of variance).
6-1 ANOVA: Using Statistics
• In an analysis of variance:We have r independent random samples, each one
corresponding to a population subject to a different treatment.
We have: n = n1+ n2+ n3+ ...+nr total observations.
r sample means: x1, x2 , x3 , ... , xr These r sample means can be used to calculate an estimator
of the population variance. If the population means are equal, we expect the variance among the sample means to be small.
r sample variances: s12, s2
2, s32, ...,sr
2
These sample variances can be used to find a pooled estimator of the population variance.
Analysis of Variance: Using Statistics (continued)
• We assume independent random sampling from each of the r populations
• We assume that the r populations under study: are normally distributed, with means mi that may or may not be equal,
but with equal variances, i2.
1 2 3
Population 1 Population 2 Population 3
Analysis of Variance: Assumptions
The test statistic of analysis of variance:F(r-1, n-r) = Estimate of variance based on means from r samples
Estimate of variance based on all sample observations
That is, the test statistic in an analysis of variance is based on the ratio of two estimators of a population variance, and is therefore based on the F distribution, with (r-1) degrees of freedom in the numerator and (n-r) degrees of freedom in the denominator.
The hypothesis test of analysis of variance:
H0: 1 = 2 = 3 = 4 = ... r
H1: Not all mi (i = 1, ..., r) are equal.
6-2 The Hypothesis Test of Analysis of Variance
x
x
x
When the null hypothesis is true:
We would expect the sample means to be nearly equal, as in this illustration. And we would expect the variation among the sample means (between sample) to be small, relative to the variation found around the individual sample means (within sample).
If the null hypothesis is true, the numerator in the test statistic is expected to be small, relative to the denominator:
F(r-1, n-r)=Estimate of variance based on means from r samples
Estimate of variance based on all sample observations
When the Null Hypothesis Is True
x xx
When the null hypothesis is false: is equal to but not to , is equal to but not to , is equal to but not to , or , , and are all unequal.
In any of these situations, we would not expect the sample means to all be nearly equal. We would expect the variation among the sample means (between sample) to be large, relative to the variation around the individual sample means (within sample).
If the null hypothesis is false, the numerator in the test statistic is expected to be large, relative to the denominator:
F(r-1, n-r)=Estimate of variance based on means from r samples
Estimate of variance based on all sample observations
When the Null Hypothesis Is False
• Suppose we have 4 populations, from each of which we draw an independent random sample, with n1 + n2 + n3 + n4 = 54. Then our test statistic is:
F(4-1, 54-4)= F(3,50) = Estimate of variance based on means from 4 samples Estimate of
variance based on all 54 sample observations
543210
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0F(3,50)
f (F)
F Distribution with 3 and 50 Degrees of Freedom
2.79
=0.05
The nonrejection region (for a=0.05)in this instance is F £ 2.79, and the rejection region is F > 2.79. If the test statistic is less than 2.79 we would not reject the null hypothesis, and we would conclude the 4 population means are equal. If the test statistic is greater than 2.79, we would reject the null hypothesis and conclude that the four population means are not equal.
The ANOVA Test Statistic for r = 4 Populations and n = 54 Total Sample Observations
Randomly chosen groups of customers were served different types of coffee and asked to rate the coffee on a scale of 0 to 100: 21 were served pure Brazilian coffee, 20 were served pure Colombian coffee, and 22 were served pure African-grown coffee.
The resulting test statistic was F = 2.02
H0 1 2 3H1 Not all three means equal
n1 = 21 n2 = 20 n3 = 22 n = 21 + 20 + 22 = 63
r = 3
The critical point for = 0.05 is:
- ,( - )
H0 cannot be rejected, and we cannot conclude that any of the
population means differs significantly from the others.
:
:
, , .
. , .
F r n r F F
F F
1 3 1 63 3 2 60 3 15
2 02 2 60 3 15
543210
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0F
f( F)
F Distribution with 2 and 60 Degrees of Freedom
=0.05
Test Statistic=2.02 F(2,60)=3.15
Example 6-1
The grand mean, x, is the mean of all n = n1+ n2+ n3+...+ nr observations in all r samples.
The mean of sample i (i = 1,2,3, . . . , r):
=
xij
The grand mean, the mean of all data points:
=
xij=
where xij is the particular data point in position j within the sample from population i.
The subscript i denotes the population, or treatment, and runs from 1 to r. The subscript j
denotes the data point within the sample from population i; thus, j runs from 1 to n j
i
xij
ni
ni
xii
r
j
ni
n
i
n
n xir
1
1 1 1
.
6-3 The Theory and Computations of ANOVA: The Grand Mean
Using the Grand Mean: Table 6-1
Distance from data point to its sample mean
Distance from sample mean to grand mean
1050
x3=2
x2=11.5
x1=6
x=6.909
Treatment (j) Sample point(j) Value(xij)I=1 Triangle 1 4Triangle 2 5Triangle 3 7Triangle 4 8
Mean of Triangles 6I=2 Square 1 10Square 2 11Square 3 12Square 4 13
Mean of Squares 11.5I=3 Circle 1 1Circle 2 2Circle 3 3
Mean of Circles 2Grand mean of all data points 6.909
If the r population means are different (that is, at least two of the population means are not equal), then it is likely that the variation of the data points about their respective sample means (within sample variation) will be small relative to the variation of the r sample means about the grand mean (between sample variation).
We define an as the difference between a data pointand its sample mean. Errors are denoted by , and we have:
We define a as the deviation of a sample meanfrom the grand mean. Treatment deviations, t are given by:
i
error deviation
treatment deviation
e
e x x
t x x
ij ij i
i i
,
The ANOVA principle says:When the population means are not equal, the “average” error(within sample) is relatively small compared with the “average”treatment (between sample) deviation.
The Theory and Computations of ANOVA: Error Deviation and Treatment Deviation
Consider data point x24=13 from table 9-1. The mean of sample 2 is 11.5, and the grand mean is 6.909, so:e x x
t x x
Tot t e
Tot x x
24 24 2 13 11 5 1 5
2 2 11 5 6 909 4 591
24 2 24 1 5 4 591 6 091
24 24 13 6 909 6 091
. .
. . .
. . .
. .
or
1050
x2=11.5
x=6.909
x24=13
Total deviation:Tot24=x24-x=6.091
Treatment deviation:t2=x2-x=4.591
Error deviation:e24=x24-x2=1.5
The total deviation (Totij) is the difference between a data point (xij) and the grand mean (x):
Totij=xij - x
For any data point xij:
Tot = t + e
That is:
Total Deviation = Treatment Deviation + Error Deviation
The Theory and Computations of ANOVA: The Total Deviation
Total Deviation = Treatment Deviation + Error Deviation
Squared Deviations
The total deviation is the sum of the treatment deviation and the error deviation: + = ( ) ( )
Notice that the sample mean term ( ) cancels out in the above addition, which
simplifies the equation.
2 +
2= ( )
2( )
2
ti
eij
xi
x xij xi
xij x Totij
xi
ti
eij
xi
x xij xi
Totij xij x
( )
( )2 2
The Theory and Computations of ANOVA: Squared Deviations
Sums of Squared Deviations
2
+2
= ni( )2 ( )2
SST = SSTR + SSE
Totijj
nj
i
rnitii
reijj
nj
i
r
xij
xj
nj
i
rxi
xi
rxij
xij
nj
i
r
2
11 1 11
2
11 1 11
( )
The Sum of Squares PrincipleThe total sum of squares (SST) is the sum of two terms: the sum of squares for treatment (SSTR) and the sum of squares for error (SSE).
SST = SSTR + SSE
The Theory and Computations of ANOVA: The Sum of Squares Principle
SST
SSTR SSTE
SST measures the total variation in the data set, the variation of all individual data points from the grand mean.
SSTR measures the explained variation, the variation of individual sample means from the grand mean. It is that part of the variation that is possibly expected, or explained, because the data points are drawn from different populations. It’s the variation between groups of data points.
SSE measures unexplained variation, the variation within each group that cannot be explained by possible differences between the groups.
The Theory and Computations of ANOVA: Picturing The Sum of Squares Principle
The number of degrees of freedom associated with SST is (n - 1).n total observations in all r groups, less one degree of freedom lost with the calculation of the grand mean
The number of degrees of freedom associated with SSTR is (r - 1).r sample means, less one degree of freedom lost with thecalculation of the grand mean
The number of degrees of freedom associated with SSE is (n-r). n total observations in all groups, less one degree of freedomlost with the calculation of the sample mean from each of r groups
The degrees of freedom are additive in the same way as are the sums of squares: df(total) = df(treatment) + df(error) (n - 1) = (r - 1) + (n - r)
The Theory and Computations of ANOVA: Degrees of Freedom
Recall that the calculation of the sample variance involves the division of the sum of squared deviations from the sample mean by the number of degrees of freedom. This principle is applied as well to find the mean squared deviations within the analysis of variance.
Mean square treatment (MSTR):
Mean square error (MSE):
Mean square total (MST):
(Note that the additive properties of sums of squares do not extend to the mean squares. MST ¹ MSTR + MSE).
MSTRSSTRr
( )1
MSESSEn r
( )
MSTSSTn
( )1
The Theory and Computations of ANOVA: The Mean Squares
E MSE
E MSTRni ir
i
( )
and
( )( ) when the null hypothesis is true
> when the null hypothesis is false
where is the mean of population i and is the combined mean of all r populations.
2
22
1
2
2
That is, the expected mean square error (MSE) is simply the common population variance (remember the assumption of equal population variances), but the expected treatment sum of squares (MSTR) is the common population variance plus a term related to the variation of the individual population means around the grand population mean.
If the null hypothesis is true so that the population means are all equal, the second term in the E(MSTR) formulation is zero, and E(MSTR) is equal to the common population variance.
The Theory and Computations of ANOVA: The Expected Mean Squares
When the null hypothesis of ANOVA is true and all r population means are equal, MSTR and MSE are two independent, unbiased estimators of the common population variance 2.
On the other hand, when the null hypothesis is false, then MSTR will tend to be larger than MSE.
So the ratio of MSTR and MSE can be used as an indicator of the equality or inequality of the r population means.
This ratio (MSTR/MSE) will tend to be near to 1 if the null hypothesis is true, and greater than 1 if the null hypothesis is false. The ANOVA test, finally, is a test of whether (MSTR/MSE) is equal to, or greater than, 1.
Expected Mean Squares and the ANOVA Principle
Under the assumptions of ANOVA, the ratio (MSTR/MSE) possess an F distribution with (r-1) degrees of freedom for the numerator and (n-r) degrees of freedom for the denominator when the null hypothesis is true.
The test statistic in analysis of variance:
( - , - )FMSTRMSEr n r1
The Theory and Computations of ANOVA: The F Statistic
( )2
ni( )2
Critical point ( = 0.01): 8.65
H0
may be rejected at the 0.01 level
of significance.
SSE xij
xij
nj
i
r
SSTR xi
xi
r
MSTRSSTR
r
MSESSTR
n r
FMSTR
MSE
117
1
1159 9
1
159 9
3 179 95
17
82 125
2 8
79 95
2 12537 62
.
.
( ).
.
( , )
.
.. .
Treatment (i) i j Value (x ij ) (xij -xi ) (xij -xi )2
Triangle 1 1 4 -2 4
Triangle 1 2 5 -1 1
Triangle 1 3 7 1 1
Triangle 1 4 8 2 4
Square 2 1 10 -1.5 2.25
Square 2 2 11 -0.5 0.25Square 2 3 12 0.5 0.25Square 2 4 13 1.5 2.25
Circle 3 1 1 -1 1
Circle 3 2 2 0 0
Circle 3 3 3 1 1
73 0 17
Treatment (xi -x) (xi -x)2 ni (xi -x)2
Triangle -0.909 0.826281 3.305124
Square 4.591 21.077281 84.309124
Circle -4.909 124.098281 72.294843
159.909091
6-4 The ANOVA Table and Examples
Source ofVariation
Sum ofSquares
Degrees ofFreedom Mean Square F Ratio
Treatment SSTR=159.9 (r-1)=2 MSTR=79.95 37.62
Error SSE=17.0 (n-r)=8 MSE=2.125
Total SST=176.9 (n-1)=10 MST=17.69
100
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0F(2,8)
f(F
)
F Distribution for 2 and 8 Degrees of Freedom
8.65
0.01
Computed test statistic=37.62
The ANOVA Table summarizes the ANOVA calculations.In this instance, since the test statistic is greater than the critical point for an a=0.01 level of significance, the null hypothesis may be rejected, and we may conclude that the means for triangles, squares, and circles are not all equal.
ANOVA Table
Treat Value
1 41 51 71 82 102 112 122 133 13 23 3
MTB > Oneway 'Value' 'Treat'.
One-Way Analysis of Variance
Analysis of Variance on Value Source DF SS MS F pTreat 2 159.91 79.95 37.63 0.000Error 8 17.00 2.12Total 10 176.91
The MINITAB output includes not only the ANOVA table and the test statistic, but it also gives a p-value corresponding to the calculated F-ratio. In this instance the p-value is approximately 0, so the null hypothesis can be rejected at any common level of significance.
Using the Computer
The EXCEL output is created by selecting ANOVA: SINGLE FACTOR option from the DATA ANALYSIS toolkit. The critical F value is based on a = 0.01. The p-value is very small, so again the null hypothesis can be rejected at any common level of significance.
Anova: Single Factor
SUMMARYGroups Count Sum Average Variance
TRIANGLE 4 24 6 3.333333333SQUARE 4 46 11.5 1.666666667CIRCLE 3 6 2 1
ANOVA
Source of Variation SS df MS F P-value F critBetween Groups 159.9090909 2 79.95454545 37.62566845 8.52698E-05 8.64906724Within Groups 17 8 2.125
Total 176.9090909 10
Using the Computer
Club Med has conducted a test to determine whether its Caribbean resorts are equally well liked by vacationing club members. The analysis was based on a survey questionnaire (general satisfaction, on a scale from 0 to 100) filled out by a random sample of 40 respondents from each of 5 resorts.
Source ofVariation
Sum ofSquares
Degrees ofFreedom Mean Square F Ratio
Treatment SSTR= 14208 (r-1)= 4 MSTR= 3552 7.04
Error SSE=98356 (n-r)= 195 MSE= 504.39
Total SST=112564 (n-1)= 199 MST= 565.65
Resort Mean Response (x )i
Guadeloupe 89
Martinique 75
Eleuthra 73
Paradise Island 91
St. Lucia 85
SST=112564 SSE=98356
F(4,200)
F Distribution with 4 and 200 Degrees of Freedom
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
f(F
)
3.41
0.01
Computed test statistic=7.04
The resultant F ratio is larger than the critical point for = 0.01, so the null hypothesis may be rejected.
Example 6-2: Club Med
Source ofVariation
Sum ofSquares
Degrees ofFreedom Mean Square F Ratio
Treatment SSTR= 879.3 (r-1)=3 MSTR= 293.1 8.52
Error SSE= 18541.6 (n-r)= 539 MSE=34.4
Total SST= 19420.9 (n-1)=542 MST= 35.83
Given the total number of observations (n = 543), the number of groups (r = 4), the MSE (34. 4), and the F ratio (8.52), the remainder of the ANOVA table can be completed. The critical point of the F distribution for = 0.01 and (3, 400) degrees of freedom is 3.83. The test statistic in this example is much larger than this critical point, so the p value associated with this test statistic is less than 0.01, and the null hypothesis may be rejected.
Example 6-3: Job Involvement
Anova: Single Factor
SUMMARYGroups Count Sum Average Variance
Michael 21 1979 94.23809524 8.59047619Damon 21 1644 78.28571429 41.11428571Allen 21 1381 65.76190476 352.8904762
ANOVASource of Variation SS df MS F P-value F crit
Between Groups 8555.52381 2 4277.761905 31.87639718 3.69732E-10 3.150411487Within Groups 8051.904762 60 134.1984127
Total 16607.42857 62
The test statistic value is 31.8764, way over the critical point for F(2, 60) of 3.15 when = 0.05.
The GM should do whatever it takes to sign Michael.
See text for data and information on the problem
Example 6-4: NBA Franchise
Data ANOVADo Not Reject H0 Stop
Reject H0
The sample means are unbiased estimators of the population means.
The mean square error (MSE) is an unbiased estimator of the common population variance.
Further Analysis
Confidence Intervals for Population Means
Tukey Pairwise Comparisons Test
The ANOVA Diagram
6-5 Further Analysis
A (1 - ) 100% confidence interval for , the mean of population i: i
where t is the value of the distribution with ) degrees of
freedom that cuts off a right - tailed area of2
.2
x tMSEni
i
2
t n - r
Confidence Intervals for Population Means
x tMSEn
x xi
i
i i
2
1 96504 39
406 96
89 6 96 82 04 95 96]75 6 96 68 04 81 96]73 6 96 66 04 79 96]91 6 96 84 04 97 96]85 6 96 78 04 91 96]
..
.
. [ . , .
. [ . , .
. [ . , .
. [ . , .
. [ . , .
Resort Mean Response (x i)
Guadeloupe 89
Martinique 75
Eleuthra 73
Paradise Island 91
St. Lucia 85
SST = 112564 SSE = 98356
ni = 40 n = (5)(40) = 200
MSE = 504.39
Confidence Intervals for Population Means
The Tukey Pairwise Comparison test, or Honestly Significant Differences (MSD) test, allows us to compare every pair of population means with a single level of significance.
It is based on the studentized range distribution, q, with r and (n-r) degrees of freedom.
The critical point in a Tukey Pairwise Comparisons test is the Tukey Criterion:
where ni is the smallest of the r sample sizes.
The test statistic is the absolute value of the difference between the appropriate sample means, and the null hypothesis is rejected if the test statistic is greater than the critical point of the Tukey Criterion
T qMSEni
Note that there are r
2 pairs of population means to compare. For example, if = :
H 0 H 0 H 0
H1 H1 H1
r
rr
!
!( ) !
: : :
: : :
2 23
1 2 1 3 2 3
1 2 1 3 2 3
The Tukey Pairwise Comparison Test
The test statistic for each pairwise test is the absolute difference between the appropriate sample means. i Resort Mean I. H0: 1 2 VI. H0: 2 4
1 Guadeloupe 89 H1: 1 2 H1: 2 4
2 Martinique 75 |89-75|=14>13.7* |75-91|=16>13.7*3 Eleuthra 73 II. H0: 1 3 VII. H0: 2 5
4 Paradise Is. 91 H1: 1 3 H1: 2 5
5 St. Lucia 85 |89-73|=16>13.7* |75-85|=10<13.7 III. H0: 1 4 VIII. H0: 3 4
The critical point T0.05 for H1: 1 4 H1: 3 4
r=5 and (n-r)=195 |89-91|=2<13.7 |73-91|=18>13.7*degrees of freedom is: IV. H0: 1 5 IX. H0: 3 5
H1: 1 5 H1: 3 5
|89-85|=4<13.7 |73-85|=12<13.7 V. H0: 2 3 X. H0: 4 5
H1: 2 3 H1: 4 5
|75-73|=2<13.7 |91-85|= 6<13.7Reject the null hypothesis if the absolute value of the difference between the sample means is greater than the critical value of T. (The hypotheses marked with * are rejected.)
T qMSEni
3 86504 4
4013 7.
..
The Tukey Pairwise Comparison Test: The Club Med Example
We rejected the null hypothesis which compared the means of populations 1 and 2, 1 and 3, 2 and 4, and 3 and 4. On the other hand, we accepted the null hypotheses of the equality of the means of populations 1 and 4, 1 and 5, 2 and 3, 2 and 5, 3 and 5, and 4 and 5.
The bars indicate the three groupings of populations with possibly equal means: 2 and 3; 2, 3, and 5; and 1, 4, and 5.
123 45
Picturing the Results of a Tukey Pairwise Comparisons Test: The Club Med Example
• A statistical model is a set of equations and assumptions that capture the essential characteristics of a real-world situation The one-factor ANOVA model:
xij=i+ij=+i+ij
where eij is the error associated with the jth member of the ith population. The errors are assumed to be normally distributed with mean 0 and variance 2.
• A factor is a set of populations or treatments of a single kind. For example: One factor models based on sets of resorts, types of airplanes, or kinds of sweaters Two factor models based on firm and location Three factor models based on color and shape and size of an ad.
• Fixed-Effects and Random Effects A fixed-effects model is one in which the levels of the factor under study (the
treatments) are fixed in advance. Inference is valid only for the levels under study. A random-effects model is one in which the levels of the factor under study are
randomly chosen from an entire population of levels (treatments). Inference is valid for the entire population of levels.
6-6 Models, Factors and Designs
• A completely-randomized design is one in which the elements are assigned to treatments completely at random. That is, any element chosen for the study has an equal chance of being assigned to any treatment.
• In a blocking design, elements are assigned to treatments after first being collected into homogeneous groups. In a completely randomized block design, all members of
each block (homogeneous group) are randomly assigned to the treatment levels.
In a repeated measures design, each member of each block is assigned to all treatment levels.
Experimental Design
• In a two-way ANOVA, the effects of two factors or treatments can be investigated simultaneously. Two-way ANOVA also permits the investigation of the effects of either factor alone and of the two factors together. The effect on the population mean that can be attributed to the levels of either
factor alone is called a main effect. An interaction effect between two factors occurs if the total effect at some pair
of levels of the two factors or treatments differs significantly from the simple addition of the two main effects. Factors that do not interact are called additive.
• Three questions answerable by two-way ANOVA: Are there any factor A main effects? Are there any factor B main effects? Are there any interaction effects between factors A and B?
• For example, we might investigate the effects on vacationers’ ratings of resorts by looking at five different resorts (factor A) and four different resort attributes (factor B). In addition to the five main factor A treatment levels and the four main factor B treatment levels, there are (5*4=20) interaction treatment levels.3
6-7 Two-Way Analysis of Variance
• xijk=+i+ j + (ijk + ijk
where is the overall mean;
i is the effect of level i(i=1,...,a) of factor A;
j is the effect of level j(j=1,...,b) of factor B;
jj is the interaction effect of levels i and j;
jjk is the error associated with the kth data point from level i of factor A and level j of factor B.
jjk is assumed to be distributed normally with mean zero and variance 2 for all i, j, and k.
The Two-Way ANOVA Model
Guadeloupe Martinique EleuthraParadiseIsland St. Lucia
Friendship n11 n21 n31 n41 n51
Sports n12 n22 n32 n42 n52
Culture n13 n23 n33 n43 n53
Excitement n14 n24 n34 n44 n54
Factor A: Resort
Fac
tor
B:
Attr
ibut
e
Resort
Ra
tin
g
Graphical Display of Effects
EleuthraMartinique
St. LuciaGuadeloupe
Paradise island
Friendship
ExcitementSportsCulture
Eleuthra/sports interaction: Combined effect greater than additive main effects
Sports
Friendship
Attribute
Resort
Excitement
Culture
Rating
EleuthraMartinique
St. LuciaGuadeloupe
Paradise Island
Two-Way ANOVA Data Layout: Club Med Example
• Factor A main effects test:H0: i=0 for all i=1,2,...,aH1: Not all i are 0
• Factor B main effects test:H0: j=0 for all j=1,2,...,bH1: Not all i are 0
• Test for (AB) interactions:H0: ij=0 for all i=1,2,...,a and j=1,2,...,bH1: Not all ij are 0
Hypothesis Tests a Two-Way ANOVA
In a two-way ANOVA: xijk=+i+ j + (ijk + ijk
SST = SSTR +SSE SST = SSA + SSB +SS(AB)+SSE
SST SSTR SSE
x x x x x x
SSTR SSA SSB SS AB
xix x
jx x
ijxixjx
( ) ( ) ( )
( )
( ) ( ) ( )
2 2 2
2 2 2
Sums of Squares
Source ofVariation
Sum of Squares
Degreesof Freedom Mean Square F Ratio
Factor A SSA a-1MSA
SSAa
1
FMSAMSE
Factor B SSB b-1MSB
SSBb
1
FMSBMSE
Interaction SS(AB) (a-1)(b-1)MS AB
SS ABa b
( )( )
( )( ) 1 1
FMS ABMSE
( )
Error SSE ab(n-1)MSE
SSEab n
( )1Total SST abn-1
A Main Effect Test: F(a-1,ab(n-1)) B Main Effect Test: F(b-1,ab(n-1))
(AB) Interaction Effect Test: F((a-1)(b-1),ab(n-1))
The Two-Way ANOVA Table
Source ofVariation
Sum of Squares
Degreesof Freedom Mean Square F Ratio
Location 1824 2 912 8.94 *
Artist 2230 2 1115 10.93 *
Interaction 804 4 201 1.97
Error 8262 81 102
Total 13120 89
=0.01, F(2,81)=4.88 Both main effect null hypotheses are rejected.=0.05, F(2,81)=2.48 Interaction effect null hypotheses are not rejected.
Example 6-4: Two-Way ANOVA (Location and Artist)
6543210
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0F
f(F)
F Distribution with 2 and 81 Degrees of Freedom
F0.01=4.88
=0.01
Location test statistic=8.94Artist test statistic=10.93
6543210
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 F
f(F)
F Distribution with 4 and 81 Degrees of Freedom
Interaction test statistic=1.97
=0.05
F0.05=2.48
Hypothesis Tests
Kimball’s Inequality gives an upper limit on the true probability of at least one Type I error in the three tests of a two-way analysis:
1- (1-1) (1-2) (1-3)
Tukey Criterion for factor A:
where the degrees of freedom of the q distribution are now a and ab(n-1). Note that MSE is divided by bn.
T qMSE
bn
Overall Significance Level and Tukey Method for Two-Way ANOVA
Source ofVariation
Sum of Squares
Degreesof Freedom Mean Square F Ratio
Factor A SSA a-1 MSASSAa
1 FMSAMSE
Factor B SSB b-1MSB
SSBb
1F
MSBMSE
Factor C SSC c-1MSC
SSCc
1
FMSCMSE
Interaction (AB)
SS(AB) (a-1)(b-1)MS AB
SS ABa b
( )( )
( )( )
1 1F
MS ABMSE
( )
Interaction (AC)
SS(AC) (a-1)(c-1)MS AC
SS ACa c
( )( )
( )( ) 1 1
FMS ACMSE
( )
Interaction (BC)
SS(BC) (b-1)(c-1) MS BCSS BCb c
( )( )
( )( ) 1 1
FMS BCMSE
( )
Interaction (ABC)
SS(ABC) (a-1)(b-1)(c-1) MS ABCSS ABC
a b c( )
( )( )( )( )
1 1 1F
MS ABCMSE
( )
Error SSE abc(n-1) MSESSE
abc n ( )1
Total SST abcn-1
Three-Way ANOVA Table
• A block is a homogeneous set of subjects, grouped to minimize within-group differences.
• A competely-randomized design is one in which the elements are assigned to treatments completely at random. That is, any element chosen for the study has an equal chance of being assigned to any treatment.
• In a blocking design, elements are assigned to treatments after
first being collected into homogeneous groups. – In a completely randomized block design, all members of
each block (homogenous group) are randomly assigned to the treatment levels.
– In a repeated measures design, each member of each block is assigned to all treatment levels.
6-8 Blocking Designs
Source of Variation Sum of Squares df Mean Square F RatioBlocks 2750 39 70.51 0.69Treatments 2640 2 1320 12.93Error 7960 78 102.05Total 13350 119
= 0.01, F(2, 78) = 4.88
Source of Variation Sum of Squares Degress of Freedom Mean Square F Ratio
Blocks SSBL n - 1 MSBL = SSBL/(n-1) F = MSBL/MSETreatments SSTR r - 1 MSTR = SSTR/(r-1) F = MSTR/MSEError SSE (n -1)(r - 1)Total SST nr - 1
ANOVA Table for Blocking Designs: Example 6-5
MSE = SSE/(n-1)(r-1)
MTB > ONEWAY C1, C2;SUBC> TUKEY 0.05.One-Way Analysis of Variance
Analysis of Variance on C1 Source DF SS MS F pMethod 2 1348.45 674.23 69.42 0.000Error 63 611.91 9.71Total 65 1960.36 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+---------+---------+---------+-- 1 22 22.773 3.131 (--*--) 2 22 30.636 3.824 (---*--) 3 22 19.955 2.171 (--*--) ----+---------+---------+---------+--Pooled StDev = 3.117 20.0 24.0 28.0 32.0
Tukey's pairwise comparisons Family error rate = 0.0500Individual error rate = 0.0193
Critical value = 3.39
Intervals for (column level mean) - (row level mean) 1 2 2 -10.116 -5.611
3 0.566 8.429 5.071 12.934
Using the Computer
Anova: Single Factor
SUMMARYGroups Count Sum Average Variance
METHOD A 22 501 22.77272727 9.803030303METHOD B 22 674 30.63636364 14.62337662METHOD C 22 439 19.95454545 4.712121212
ANOVASource of Variation SS df MS F P-value F crit
Between Groups 1348.454545 2 674.2272727 69.41606002 1.18041E-16 3.14280868Within Groups 611.9090909 63 9.712842713
Total 1960.363636 65
95% Confidence MethodIntervals on Means A B C
LOWER BOUND 21.44493 29.309 18.62675UPPER BOUND 24.10052 31.964 21.28234
Using the Computer: Example 6-6 Using Excel
• Introduction.
• The Multivariate Normal Distribution.
• Discriminant Analysis.
• Principal Components and Factor Analysis.
• Using the Computer.
• Summary and Review of Terms.
6-9 Multivariate Analysis
• A k-dimensional (vector) random variable X:X = (X1, X2, X3..., Xk)
• A realization of a k-dimensional random variable X:x = (x1, x2, x3..., xk)
• A joint cumulative probability distribution function of a k-dimensional random variable X:F(x1, x2, x3..., xk) = P(X1x1, X2x2,..., Xkxk)
6-10 The Multivariate Normal Distribution
A multivariate normal random variable has the following probability density function:
where X is the vector random variable, the term = ( is the vector of means of the component variables X and is the variance - covariance matrix. The operations ' and aretransposition and inversion of matrices, respectively, and denotes the determinant of a matrix.
1 2 k
i-1
f x x xk eX X
k( , , , )( ) ( )
, , , ),
1 21
2
12
212
1
The Multivariate Normal Distribution
f(x1,x2)
x1
x2
Picturing the Bivariate Normal Distribution
X2
X1
Group 1
Group 2
1
2
Line L
In a discriminant analysis, observations are classified into two or more groups, depending on the value of a multivariate discriminant function.
As the figure illustrates, it may be easier to classify observations by looking at them from another direction. The groups appear more separated when viewed from a point perpendicular to Line L, rather than from a point perpendicular to the X1 or X2 axis. The discriminant function gives the direction that maximizes the separation between the groups.
6-11 Discriminant Analysis
Group 1 Group 2
CCutting Score
The form of the estimated predicted equation:D= b0 +b1X1+b2X2+...+bkXk
where the bi are the discriminant weights. b0 is a constant.
The intersection of the normal marginal distributions of two groups gives the cutting score, which is used to assign observations to groups. Observations with scores less than C are assigned to group 1, and observations with scores greater than C are assigned to group 2. Since the distributions may overlap, some observations may be misclassified.
The model may be evaluated in terms of the percentages of observations assigned correctly and incorrectly.
The Discriminant Function
Discriminant 'Repay' 'Assets' 'Debt' 'Famsize'.Group 0 1Count 14 18
Summary of ClassificationPut into ....True Group....Group 0 10 10 51 4 13Total N 14 18N Correct 10 13Proport. 0.714 0.722
N = 32 N Correct = 23 Prop. Correct = 0.719
Linear Discriminant Function for Group 0 1Constant -7.0443 -5.4077Assets 0.0019 0.0548Debt 0.0758 0.0113Famsize 3.5833 2.8570
Discriminant Analysis: Example 6-7 (Minitab)
Summary of Misclassified ObservationsObservation True Pred Group Sqrd Distnc Probability Group Group 4 ** 1 0 0 6.966 0.515 1 7.083 0.485 7 ** 1 0 0 0.9790 0.599 1 1.7780 0.401 21 ** 0 1 0 2.940 0.348 1 1.681 0.652 22 ** 1 0 0 0.3812 0.775 1 2.8539 0.225 24 ** 0 1 0 5.371 0.454 1 5.002 0.546 27 ** 0 1 0 2.617 0.370 1 1.551 0.630 28 ** 1 0 0 1.250 0.656 1 2.542 0.344 29 ** 1 0 0 1.703 0.782 1 4.259 0.218 32 ** 0 1 0 1.84529 0.288 1 0.03091 0.712
Example 6-7: Misclassified Observations
1 0 set width 80 2 data list free / assets income debt famsize job repay 3 begin data 35 end data 36 discriminant groups = repay(0,1) 37 /variables assets income debt famsize job 38 /method = wilks 39 /fin = 1 40 /fout = 1 41 /plot 42 /statistics = all Number of cases by group Number of cases REPAY Unweighted Weighted Label 0 14 14.0 1 18 18.0 Total 32 32.0
Example 6-7: SPSS Output (1)
- - - - - - - - D I S C R I M I N A N T A N A L Y S I S - - - - - - - -On groups defined by REPAY Analysis number 1 Stepwise variable selection Selection rule: minimize Wilks' Lambda Maximum number of steps.................. 10 Minimum tolerance level.................. .00100 Minimum F to enter....................… 1.00000 Maximum F to remove...................... 1.00000 Canonical Discriminant Functions Maximum number of functions.............. 1 Minimum cumulative percent of variance... 100.00 Maximum significance of Wilks' Lambda.... 1.0000 Prior probability for each group is .50000
Example 6-7: SPSS Output (2)
---------------- Variables not in the Analysis after Step 0 ---------------- MinimumVariable Tolerance Tolerance F to Enter Wilks' Lambda ASSETS 1.0000000 1.0000000 6.6151550 .8193329INCOME 1.0000000 1.0000000 3.0672181 .9072429DEBT 1.0000000 1.0000000 5.2263180 .8516360FAMSIZE 1.0000000 1.0000000 2.5291715 .9222491JOB 1.0000000 1.0000000 .2445652 . 9919137 * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * At step 1, ASSETS was included in the analysis. Degrees of Freedom Signif. Between GroupsWilks' Lambda .81933 1 1 30.0Equivalent F 6.61516 1 30.0 .0153
Example 6-7: SPSS Output (3)
---------------- Variables in the Analysis after Step 1 ----------------Variable Tolerance F to Remove Wilks' LambdaASSETS 1.0000000 6.6152 ---------------- Variables not in the Analysis after Step 1 ------------ MinimumVariable Tolerance Tolerance F to Enter Wilks' Lambda INCOME .5784563 .5784563 . 0090821 .8190764DEBT .9706667 .9706667 6.0661878 .6775944FAMSIZE .9492947 .9492947 3.9269288 .7216177JOB .9631433 .9631433 .0000005 .8193329 At step 2, DEBT was included in the analysis. Degrees of Freedom Signif. Between GroupsWilks' Lambda .67759 2 1 30.0Equivalent F 6.89923 2 29.0 .0035
Example 6-7: SPSS Output (4)
----------------- Variables in the Analysis after Step 2 ---------------- Variable Tolerance F to Remove Wilks' LambdaASSETS .9706667 7.4487 .8516360DEBT .9706667 6.0662 .8193329 -------------- Variables not in the Analysis after Step 2 ------------- MinimumVariable Tolerance Tolerance F to Enter Wilks' LambdaINCOME .5728383 .5568120 .0175244 .6771706FAMSIZE .9323959 .9308959 2.2214373 .6277876JOB .9105435 .9105435 .2791429 .6709059 At step 3, FAMSIZE was included in the analysis. Degrees of Freedom Signif. Between GroupsWilks' Lambda .62779 3 1 30.0Equivalent F 5.53369 3 28.0 .0041
Example 6-7: SPSS Output (5)
------------- Variables in the Analysis after Step 3 ----------------Variable Tolerance F to Remove Wilks' LambdaASSETS .9308959 8.4282 .8167558DEBT .9533874 4.1849 .7216177FAMSIZE .9323959 2.2214 .6775944 ------------- Variables not in the Analysis after Step 3 ------------ MinimumVariable Tolerance Tolerance F to Enter Wilks' LambdaINCOME .5725772 .5410775 .0240984 .6272278JOB .8333526 .8333526 .0086952 .6275855 Summary Table Action Vars Wilks'Step Entered Removed in Lambda Sig. Label 1 ASSETS 1 .81933 .0153 2 DEBT 2 .67759 .0035 3 FAMSIZE 3 .62779 .0041
Example 6-7: SPSS Output (6)
Classification function coefficients(Fisher's linear discriminant functions) REPAY = 0 1 ASSETS .0018509 .0547891DEBT .0758239 .0113348FAMSIZE 3.5833063 2.8570101(Constant) -7.7374079 -6.1008660
Unstandardized canonical discriminant function coefficients Func 1 ASSETS -.0352245DEBT .0429103FAMSIZE .4832695(Constant) -.9950070
Example 6-7: SPSS Output (7)
Case Mis Actual Highest Probability 2nd Highest DiscrimNumber Val Sel Group Group P(D/G) P(G/D) Group P(G/D) Scores 1 1 1 .1798 .9587 0 .0413 -1.9990 2 1 1 .3357 .9293 0 .0707 -1.6202 3 1 1 .8840 .7939 0 .2061 -.8034 4 1 ** 0 .4761 .5146 1 .4854 .1328 5 1 1 .3368 .9291 0 .0709 -1.6181 6 1 1 .5571 .5614 0 .4386 -.0704 7 1 ** 0 .6272 .5986 1 .4014 .3598 8 1 1 .7236 .6452 0 .3548 -.3039 ........................................................................... 20 0 0 .1122 .9712 1 .0288 2.4338 21 0 ** 1 .7395 .6524 0 .3476 -.3250 22 1 ** 0 .9432 .7749 1 .2251 .9166 23 1 1 .7819 .6711 0 .3289 -.3807 24 0 ** 1 .5294 .5459 0 .4541 -.0286 25 1 1 .5673 .8796 0 .1204 -1.2296 26 1 1 .1964 .9557 0 .0443 -1.9494 27 0 ** 1 .6916 .6302 0 .3698 -.2608 28 1 ** 0 .7479 .6562 1 .3438 .5240 29 1 ** 0 .9211 .7822 1 .2178 .9445 30 1 1 .4276 .9107 0 .0893 -1.4509 31 1 1 .8188 .8136 0 .1864 -.8866 32 0 ** 1 .8825 .7124 0 .2876 -.5097
Example 6-7: SPSS Output (8)
Classification results - No. of Predicted Group Membership Actual Group Cases 0 1-------------------- ------ -------- -------- Group 0 14 10 4 71.4% 28.6% Group 1 18 5 13 27.8% 72.2% Percent of "grouped" cases correctly classified: 71.88%
Example 6-7: SPSS Output (9)
All-groups Stacked Histogram Canonical Discriminant Function 1 4 + + |
| |
|F | |r 3 + 2
+e | 2
|q | 2
| u | 2
|e 2 + 2 1 2 +n | 2 1 2 |c | 2 1 2 |y | 2 1 2 | 1 + 22 222 2 222 121 212112211 2 1 11 1 1 1 + | 22 222 2 222 121 212112211 2 1 11 1 1 1 | | 22 222 2 222 121 212112211 2 1 11 1 1 1 | | 22 222 2 222 121 212112211 2 1 11 1 1 1 | X---------------------+---------------------+---------------------+---------------------+---------------------+---------------------X out -2.0 -1.0 .0 1.0 2.0 out Class 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Centroids 2 1
Example 6-7: SPSS Output (10)
First Component
Second Component
x
y
Total Variance
VarianceRemaining After
Extraction of
First Second Third
Component
6-12 Principal Components and Factor Analysis
The k original Xi variables written as linear combinations of a smaller set of m common factors and a unique component for each variable:
X1 = b11F1+ b12F2 +...+ b1mFm + U1
X1 = b21F1+ b22F2 +...+ b2mFm + U2 . . .
Xk = bk1F1+ bk2F2 +...+ bkmFm + Uk
The Fj are the common factors. Each Ui is the unique component of variable Xi. The coefficients bij are called the factor loadings.
Total variance in the data is decomposed into the communality, the common factor component, and the specific part.
Factor Analysis
Factor 2
Factor 1
Rotated Factor 2
Rotated Factor 1
Orthogonal RotationFactor 2
Factor 1
Rotated Factor 2
Rotated Factor 1
Oblique Rotation
Rotation of Factors
Factor LoadingsSatisfaction with: 1 2 3 4 CommunalityInformation1 0.87 0.19 0.13 0.22 0.85832 0.88 0.14 0.15 0.13 0.83343 0.92 0.09 0.11 0.12 0.88104 0.65 0.29 0.31 0.15 0.6252Variety5 0.13 0.82 0.07 0.17 0.72316 0.17 0.59 0.45 0.14 0.59917 0.18 0.48 0.32 0.22 0.41368 0.11 0.75 0.02 0.12 0.58949 0.17 0.62 0.46 0.12 0.639310 0.20 0.62 0.47 0.06 0.6489Closure11 0.17 0.21 0.76 0.11 0.662712 0.12 0.10 0.71 0.12 0.5429Pay13 0.17 0.14 0.05 0.51 0.311114 0.10 0.11 0.15 0.66 0.4802
Factor Analysis of Satisfaction Items