sta305 week21 the one-factor model statistical model is used to describe data. it is an equation...
TRANSCRIPT
![Page 1: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/1.jpg)
STA305 week2 1
The One-Factor Model
• Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon the levels of the treatment factors.
• Let Yij be a random variable that represents the response obtained on the j-th observation of the i-th treatment.
• Let μ denote the overall expected response.
• The expected response for an experimental unit in the i-th treatment group is μi = μ + τi
• τi is deviation of i-th mean from overall mean; it is referred to as the effect of treatment i.
![Page 2: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/2.jpg)
STA305 week2 2
• The model is
where is the deviation of the individual’s response from the
treatment group mean.
• is known as the random or experimental error.
ijiijY
ij
ij
![Page 3: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/3.jpg)
STA305 week2 3
Fixed Effects versus Random Effects
• In some cases the treatments are specifically chosen by the experimenter from all possible treatments.
• The conclusions drawn from such an experiment apply only to these treatments and cannot be generalized to other treatments not included in experiment.
• This is called a fixed effects model
• In other cases, the treatments included in the experiment can be regarded as a random selection from the set of all possible treatments.
• In this situation, conclusions based on the experiment can be generalized to other treatments.
• When the treatments are random sample, treatment effects, τi are random variables.
• This model is called a random effects model or a components of variance model.
• The random effects model will be studied after the fixed effects model
![Page 4: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/4.jpg)
STA305 week2 4
More about the Fixed Effects Model
• As specified in slide (2) the model is
Where are i.i.d. with distribution N(0, σ2)
• It follows that response of experimental unit j in treatment group i, Yij , is normally distributed with
• In other words
ijiijY
iijYE
2 ijij VarYVar
2,~ iij NY
ij
![Page 5: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/5.jpg)
STA305 week2 5
Treatment Effects
• Recall that treatment effects have been defined as deviations from overall mean, and so the model can be parameterized so that:
• In the special case where r1 = r2 = · · · = ra = r this condition reduces to
• The hypothesis that there is no treatment effect can be expressed mathematically as:
H0 : μ1 = μ2 = · · · = μa
Ha : not all μi are equal
• This can be expressed equivalently in terms of the τi:
H0 : τ1 = τ2 = · · · = τa = 0
Ha : not all τi are equal to 0
a
iiir
1
0
a
ii
1
0
![Page 6: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/6.jpg)
STA305 week2 6
’Dot’ Notation
• “Dot” notation will be used to denote treatment and overall totals, as well as treatment and overall means.
• The sum of all observations in the i-th treatment group will be denoted as
• Similarly, the sum of all responses in all treatment groups is denoted:
• The treatment and overall means are:
i
i
irii
r
jiji YYYYY
21
1
a
i
r
jij
i
YY1 1
i
ir
jij
ii r
YY
rY
i
1
1
a
i
r
jij n
YY
nY
i
1 1
1
![Page 7: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/7.jpg)
Rationale for Analysis of Variance
• Consider all of the data from the a treatment groups as a whole.
• The variability in the data may come from two sources:
1) treatment means differ from overall mean, this is called between
group variability.
2) within a given treatment group individual observations differ
from group mean, this is called within group variability.
STA305 week2 7
![Page 8: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/8.jpg)
Total Sum of Squares
• Total variation in data set as a whole is measured by the total sum of squares. It is given by
• Each deviation from the overall sample mean can be expressed as the sum of 2 parts:
1) deviation of the observation from the group mean.
2) deviation of the group mean from the overall mean
• In other words…
• The SST can then be written as…
STA305 week2 8
a
i
r
jijT
i
YYSS1 1
2
![Page 9: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/9.jpg)
Expected Sums of Squares
• Finding the expected value of the sums of squares for error and treatment will lead us to a test of the hypothesis of no treatment effect, i.e., H0 : τ1 = τ2 = · · · = τa = 0
• We start by finding the expected value of SSE….
• We continue with the expected value of SSTreat
STA305 week2 9
![Page 10: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/10.jpg)
Mean Squares
• As we have seen in the calculation above, the MSE = SSE/(n − a) is an unbiased estimator of σ2.
• The MSE is called the mean square for error.
• The degrees of freedom associated with SSE are n − a and it follows
that E(MSE) = σ2.
• The mean square for treatment is defined to be:
MSTreat = SSTreat / (a-1).
• The expected value of MSTreat is
STA305 week2 10
a
iiiTreat r
aMSE
1
22
1
1
![Page 11: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/11.jpg)
Hypothesis Testing
• Recall that our goal is to test whether there is a treatment effect.
• The hypothesis of interest is
H0 : τ1 = τ2 = · · · = τa = 0
Ha : not all τi are equal to 0
• Notice that if H0 is true, then
• On the other hand, if H0 is false, then at least one τa ≠ 0, in which case
and so E (MSTreat) > E (MSE)
• On average, then, the ratio MSTreat/MSE should be small if H0 is true, and large otherwise.
• We use this to develop formal test.STA305 week2 11
E
a
ii
a
iiiTreat MSEr
ar
aMSE
2
1
22
1
22 01
1
1
1
02iir
![Page 12: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/12.jpg)
Cochran’s Theorem
• Let Z1,Z2, . . . ,Zn be i.i.d. N(μ, 1).
• Suppose that where Qj has d.f vj.
• A necessary and sufficient condition for the Qj to be independent of
one another, and for Qj ~ χ2(vj) is that .
• Cochran’s theorem implies that SSE/σ2 and SSTreat/ σ2 have
independent χ2 distributions with n – a and a − 1 d.f., respectively.
• Recall: If X1 and X2 are two independent random variables, each
with a χ2 distribution, then
STA305 week2 12
221
1
2s
n
ii QQZ
s
jj nv
1
2122
11 ,~/
/vvF
vX
vX
![Page 13: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/13.jpg)
Hypothesis Test for Treatment Effects
• Cochran’s theorem and the result just stated provide the tools to construct a formal hypothesis test of no treatment effects.
• The Hypothesis again are:
H0 : τ1 = τ2 = · · · = τa = 0
Ha : not all τi are equal to 0
• The Test Statistic is: Fobs = MSTreat/MSE
• Note that if H0 is true, then Fobs ~ F(a − 1, n − a).
• So the P-value = P(F(a − 1, n − a) > Fobs).
• We reject H0 in favor of Ha if P−value < α.
• Alternatively, reject H0 in favor of Ha if Fobs > Fα(a − 1, n − a), where
Fα(a − 1, n − a) is the upper 100 × α%-ile point of the F(a − 1, n − a) distribution.
STA305 week2 13
![Page 14: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/14.jpg)
Analysis of Variance Table
STA305 week2 14
• The results of the calculations and the hypothesis testing are best summarized in an analysis of variance table
• The ANOVA Table is given below
![Page 15: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/15.jpg)
Estimable Functions of Parameters
• A function of the model parameters is estimable if and only if it can be written as the expected value of a linear combination of the response variables.
• In other words, every estimable function is of the form
where the cij are constants
• It can be shown that from previous sections, μ, μi, and σ2 are estimable.
STA305 week2 15
a
i
r
jijij
i
YcE1 1
![Page 16: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/16.jpg)
Example - Effectiveness of Three Methods for Teaching a Programming Language
• A study was conducted to determine whether there is any difference in the effectiveness of 3 methods of teaching a particular programming language.
• The factor levels (treatments) are the three teaching methods:
1) on-line tutorial
2) personal attention of instructor plus hands-on experience
3) personal attention of instructor, but no hands-on experience
• Replication and Randomization: 5 volunteers were randomly allocated to each of the 3 teaching methods, for a total of 15 study participants.
• Response Variable: After the programming instruction, a test was administered to determine how well the students had learned the programming language.
• Research Question: Do the data provide any evidence that the instruction methods differ with respect to test score.
• The data and the solutions are….STA305 week2 16
![Page 17: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/17.jpg)
Conducting an ANOVA in SAS
• There are several procedures in SAS that can be used to do an analysis of variance.
• PROC GLM (for generalized linear model) will be used in this course
• To do the analysis for the Example on slide 16, start by creating a SAS dataset:
data teach ;
input method score ;
cards ;
1 73
1 77
.....
3 71
;
run ;
STA305 week2 17
![Page 18: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/18.jpg)
• Use this dataset to conduct an ANOVA using the following SAS code:
proc glm data = teach ;
class method ;
model score = method / ss3 ;
run ;
quit ;
• The output produced by this procedure is given in the next slide.
STA305 week2 18
![Page 19: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/19.jpg)
STA305 week2 19
![Page 20: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/20.jpg)
Estimating Model Parameters
• The ANOVA indicates whether there is a treatment effect, however,
it doesn’t provide any information about individual treatments or
how treatments compare with each other.
• To better understand outcome of experiment, estimating mean
response for each treatment group is useful.
• Also, it is useful to obtain an estimate of how much variability there
is within each treatment group.
• This involves estimating model parameters.
STA305 week2 20
![Page 21: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/21.jpg)
Variability
• Recall, on slides (9 and 10) we have showed that the MSE is unbiased estimator of σ2.
• Further, Cochran’s Theorem was used to show that SSE/ σ2 ~ χ2(n − a).
• We can use this result to calculate a 100 × (1 − α)% confidence interval for σ2.
• The CI is give by
where and are the upper and lower percentage
points of the χ2 distribution with n − a d.f., respectively.
STA305 week2 21
1
,1 2
2/2
2/1 n
SS
n
SS EE
an 2
2/1 an 22/
![Page 22: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/22.jpg)
Overall Mean
• As discussed in the beginning, the overall expected value is μ.
• Show that is unbiased estimator of μ…
• The variance of is σ2/n.
• So the 100 × (1 −α)% confidence interval for μ is:
• Further, a 100 × (1 −α)% confidence interval for μi is:
• It follows that is an unbiased estimator of the effect of treatment i, τi.
STA305 week2 22
Y
n
MSantY E
2/
i
Ei r
MSantY 2/
YYi
![Page 23: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/23.jpg)
Differences between Treatment Groups
• Differences between specific treatment groups will be important from researcher’s point of view.
• The expected difference in response between treatment groups i and j is: μi − μj = τi – τj.
• Since treatment groups are independent of each other, it follows that
• Therefore, a 100 × (1 −α)% confidence interval for τi – τj is:
STA305 week2 23
jijii rr
NYY11
,~ 2
ji
Ei rrMSantYY
112/
![Page 24: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/24.jpg)
Example - Methods for Teaching Programming Language Cont’d
• Back to the example of three teaching methods and their effect on programming test score.
• Based on the ANOVA developed earlier, we found significant difference between the three methods.
• Which method had the highest average?
• What is a 95% CI for mean difference in test scores for the 2 instructor-based methods?
STA305 week2 24
![Page 25: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/25.jpg)
Comparisons Among Treatment Means
• As mentioned above, ANOVA will indicate whether there is
significant effect of treatments overall it doesn’t indicate which
treatments are significantly different from each other.
• There are a number of methods available for making pairwise comparisons of treatment means.
STA305 week2 25
![Page 26: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/26.jpg)
Least Significant Difference (LSD)
• This method tests the hypothesis that all treatment pairs have the same mean against the alternative that at least one pair differs, that is the hypothesis are:
H0 : μi − μj = 0 for all i, j
Ha : μi − μj ≠ 0 for at least one pair i, j
• In testing difference between any two specific means, reject the null hypothesis if:
• In the case where the design is balanced and ri = r for all i, the condition above becomes:
STA305 week2 26
r
MSantYY E
ji
22/
ji
ji rrantYY
112/
![Page 27: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/27.jpg)
• In other words, the smallest difference between the means that would be considered statistically significant is:
• This quantity, LSD, is called the least significant difference.
• LSD method requires that the difference between each pair of means be compared to the LSD.
• In cases where difference is greater than LSD, we conclude that treatment means differ.
STA305 week2 27
r
MSantLSD E2
2/
![Page 28: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/28.jpg)
Important Notes
• As in any situation where large number of significance tests conducted, the possibility of finding large difference due to chance alone increases.
• Therefore, in case where the number of treatment groups is large, the probability of making this type of error is relatively large.
• In other words, probability of committing a Type I error will be increased above α.
• Further, although the ANOVA F-test might find a significant treatment effect, LSD method might conclude that there are no 2 treatment means that are significantly different from each other.
• This is because ANOVA F-test considers overall trend of effect of treatment on outcome, and is not restricted to pairwise comparisons.
STA305 week2 28
![Page 29: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/29.jpg)
Other Methods for Pairwise Comparisons
• Other methods for conducting pairwise comparisons are available.
• The methods that are implemented in PROC GLM in SAS include:
– Bonferonni
– Duncan’s Multiple Range Test
– Dunnett’s procedure
– Scheffe’s method
– Tukey’s test
– several otheres
• Chapter 4 of Dean & Voss discusses some of these methods.
STA305 week2 29
![Page 30: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/30.jpg)
Pairwise Comparisons in SAS
• Pairwise comparisons can be requested by including a means statement.
• The code below requests means with LSD comparison:
proc glm data = teach ;
class method ;
model score = method / ss3 ;
means method / lsd cldiff ;
run ;
• The part of the output containing the pairwise comparisons is shown in the next slide.
STA305 week2 30
![Page 31: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/31.jpg)
STA305 week2 31
![Page 32: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/32.jpg)
STA305 week2 32
![Page 33: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/33.jpg)
Contrasts
• ANOVA test indicates only whether there is an overall trend for the treatment means to differ, and does not indicate specifically which treatments are the same, which are different, etc.
• In the last few slides looked at pairwise comparisons between treatment means.
• However, comparisons that are of interest to researcher may include more then just two group. They can be linear combination of means.
STA305 week2 33
![Page 34: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/34.jpg)
Example - Does Food Decrease Effectiveness of Pain Killers?
• Researchers at pain clinic want to know whether effectiveness of two leading pain killers is same when taken on empty stomach as when taken with food.
• A study with four treatment groups was designed:
1. aspirin with no food
2. aspirin with food
3. tylenol with no food
4. tylenol with food
• In addition to determining whether there is a difference between the four treatment groups, researchers want to determine whether there is a difference between taking medication with food and taking it without.
• This second hypothesis can be expressed statistically as:
H0 : μ1 + μ3 = μ2 + μ4
Ha : μ1 + μ ≠ μ2 + μ4
STA305 week2 34
![Page 35: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/35.jpg)
• The point estimate of difference between fed and not fed conditions is based on sample means:
STA305 week2 35
4231 YYYY
![Page 36: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/36.jpg)
Hypothesis Tests Using Contrasts
• As in the example on the previous slide, the comparison of treatment means that is of interest might be a linear combination of means. That is, the hypothesis of interest would be of the form
H0 : c1μ1 + c2μ2 + · · · + caμa = 0
Ha : c1μ1 + c2μ2 + · · · + caμa ≠ 0
• The ci are constants subject to the constraints:
(i) ci > 0 for all i, and
(ii)
• Test of this hypothesis can be constructed using sample means for each treatment group.
• The linear combination c1μ1 + c2μ2 + · · · + caμa is called a contrast.
STA305 week2 36
a
i ic1
0
![Page 37: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/37.jpg)
• If the assumptions of the model are satisfied, then:
• If σ2 was known, a test of H0 could be done using:
• Since σ2 is unknown, we use its unbiased estimate, the MSE, and conduct a t-test with n − a d.f.. The test statistics is
• Recall, if X is a random variable with t(v) distribution, then X2 has F(1, v) distribution.
STA305 week2 37
a
i
a
i iiii
a
iii r
ccNYc1 1
22
1
,~
a
i ii
a
i ii
rc
Yc
1
2
1
/
a
i iiE
a
i iiobs
rcMS
Yct
1
2
1
/
![Page 38: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/38.jpg)
• So an equivalent test statistic is:
• At level α , reject H0 in favour of Ha if
Fobs > Fα(1, n − a), or equivalently if
|tobs| > tα/2 (n − a).
• The sum of squares for contrast is:
• Each contrast has 1 d.f., so the mean square for contrast is:
MScontrast = SScontrast/1
STA305 week2 38
a
i iiE
a
i iiobs
rcMS
YcF
1
2
2
1
/
a
i ii
a
i iicontrast
rc
YcSS
1
2
2
1
/
![Page 39: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/39.jpg)
Summary
• The hypothesis: H0 : c1μ1 + c2μ2 + · · · + caμa = 0
Ha : c1μ1 + c2μ2 + · · · + caμa ≠ 0
• Test Statistic
• Decision Rule: reject H0 if
Fobs > Fα(1, n − a)
STA305 week2 39
E
contrastobs MS
MSF
![Page 40: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/40.jpg)
Orthogonal Contrasts
• Very often more than one contrast will be of interest. Further, it is possible that one research question will require more than one contrast, i.e.,
H0 : μ1 = μ3 and μ2 = μ4
• Ideally, we want tests about different contrasts to be independent of each other.
• Suppose that the two contrasts of interest are:
c1μ1 + c2μ2 + · · · + caμa and d1μ1 + d2μ2 + · · · + daμa.
• These two contrasts are orthogonal to each other they iff they satisfy:
• If there are a treatments then, SSTreat can be decomposed into set of a − 1 orthogonal contrasts, each with 1 d.f. as follows
SSTreat = SScontrast1 + SScontrast2 + · · · + SScontrasta−1.
• Unless a = 2, there will be more than one set of orthogonal contrasts.
STA305 week2 40
01
a
iiidc
![Page 41: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/41.jpg)
Example - Food / Pain Killers Continued
• Refer back to the example on slide 31. The study designed with 4 treatment groups.
• The treatment sum of squares can be decomposed into 3 orthogonal contrasts.
• Since researcher interested in difference between fed & unfed, makes sense to use the following contrasts:
STA305 week2 41
![Page 42: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/42.jpg)
• Exercise: verify that each is in fact a contrast.
• Exercise: verify that contrasts are orthogonal.
• Note, there is more than one way to decompose treatment sum of squares into set of orthogonal contrasts.
• For example, instead of comparing aspirin and Tylenol, might be interested in comparing food with no food.
• In this case, compare (i) aspirin with food and Tylenol with food, (ii) aspirin without food and Tylenol without food, and (iii) the 2 food groups to the 2 no-food groups.
STA305 week2 42
![Page 43: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/43.jpg)
ANOVA Table for Orthogonal Contrasts
• Contrasts to be used in experiment must be chosen at the beginning of the study.
• The hypotheses to be tested should not be selected after viewing the data.
• Once the treatment SS has been decomposed using preplanned orthogonal contrasts, the ANOVA table can be expanded to show decomposition as shown in the next slide.
STA305 week2 43
![Page 44: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/44.jpg)
STA305 week2 44
![Page 45: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/45.jpg)
Example - Pressure on a Torsion Spring
STA305 week2 45
![Page 46: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/46.jpg)
• The figure above shows a diagram of a torsion spring.
• Pressure is applied to arms to close the spring.
• A study has been designed to examine pressure on torsion spring.
• Five different angles between arms of spring will be studied to determined their impact on the pressure: 67º, 71 º, 75 º, 79 º, and 83 º.
• Researchers are interested in whether there is an overall difference between different angle settings.
• In addition would like to study set of orthogonal contrasts which compares the 2 smallest angles to each other and 2 largest angles to each other.
• The data collected are shown in the following slide.
STA305 week2 46
![Page 47: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/47.jpg)
Torsion Spring Data
STA305 week2 47
![Page 48: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/48.jpg)
Solution
STA305 week2 48
![Page 49: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/49.jpg)
Contrasts in SAS
• To do the analysis for the last example, start by creating a SAS dataset:
data torsion ;
input angle pressure;
cards ;
67 83
67 85
71 87
71 84
...........
79 90
83 90
83 92;
run ;
STA305 week2 49
![Page 50: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/50.jpg)
• Here is an additional code that is required to specify the contrasts of interest:
proc glm data = torsion ;
class angle ;
model pressure = angle / ss3 ;
contrast ’67-71’ angle 1 -1 0 0 0 ;
contrast ’79-83’ angle 0 0 0 1 -1 ;
contrast ’sm vs lg’ angle 1 1 0 -1 -1 ;
contrast ’mid vs oth’ angle 1 1 -4 1 1 ;
run ;
quit ;
STA305 week2 50
![Page 51: STA305 week21 The One-Factor Model Statistical model is used to describe data. It is an equation that shows the dependence of the response variable upon](https://reader036.vdocument.in/reader036/viewer/2022062423/56649ebc5503460f94bc4c04/html5/thumbnails/51.jpg)
• The ANOVA part of the output is not shown here.
• The part of the output generated by the contrast statements looks like this:
Contrast DF Contrast SS Mean Square F Value Pr>F
67-71 1 3.37500000 3.37500000 2.92 0.1031
79-83 1 1.33333333 1.33333333 1.15 0.2958
sm vs lg 1 93.35294118 93.35294118 80.70 <0.0001
mid vs oth 1 0.20796354 0.20796354 0.18 0.6761
STA305 week2 51