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Chapter 12. ANOVA of Repeated Measurement Data. §1 Character of Repeated Measurement Data. Content. Data characteristic Analysis of two factors and two levels Analysis of two factors and several levels Notices.
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Chapter 12 ANOVA of Repeated Measurement Data
Content • Data characteristic • Analysis of two factors and two levels • Analysis of two factors and several levels • Notices
Objective: Inference the effects of treatment, time, and treatment*time on experimental objects Character of data: treatment factor: g （≥1 ）levels and n experimental objectives in each level, add up to gn experimental objectives. time factor: m valves at m time in each experimental objective, add up to gnm valves. Method: ANOVA
I premeasure-postmeasure design It is particular case in repeated measurement data and is also called single group premeasure-postmeasure design. g=1, m=2.
table12-1 BHP patient’s diastolic pressure in pretreatment and post-treatment（mmHg）
Compare: Table 3-3 measurement result of fat content in lactic acid (%)
Differences with paired test: 1.Two experimental units in the same pair in the paired test may be managed at random. Two experimental units can be observed at the same time. We can compare the difference of the group. The results of premeasure-postmeasure design can’t be observed at the same time although it can be arranged at pre-post experiment. But in substance it is compared with the difference of pre-post experiment. We infer the treatment to be effective on condition that we assume the time don’t affect the results.
paired/matched t-test requires the results of two experimental units in the same pair and difference to be fit for independent. The difference obeys to normal distribution. Two results of premeasure-postmeasure design are common not to independent with differences. The first time result is negative correlation to the difference in most cases. Table 12-1 as follows, the correlation of diastolic pressure before treatment with the difference is 0.602.
3. It is inferred the effective of treatment by average difference in paired design. While we can analyze average difference and analyze correlation and regression in premeasure-postmeasure design. Calculate table 12-1 as follows, the correlation coefficient of diastolic pressure in pre-post treatment is 0.963, P<0.01. We can infer the diastolic pressure after treatment by the diastolic pressure before treatment. test intercept P=0.014 ,regression coefficient P<0.01.
II premeasure-postmeasure design with contrast The diastolic pressure of HBP patients after treatment drops 16 mmHg .Although using paired test, ,it can still not be proved effective .Because the differences of rest in hospital ,surroundings, emotion can resume the diastolic pressure. So in order to prove effective, premeasure-postmeasure design should be set parallel comparison. We assign 20 light HBP patients to disposal group and comparison group at random.
Table 12-2 BHP patient’s diastolic pressure in pre-post treatment （mmHg）
III repeated measurement design When the times of repeated measurement design are over three, it is called repeated measurement design or repeated measurement data. • table12-3 Density of experimenter‘s blood glucose（mmol/L）(g=1) Test of sphericity:
The difference with randomized block design • It is to assign among the the granule (experimenters )s at random to measure in the repeated measurement design, every time point in the granule is fixed , can't assign at random , such as form 12-5, A,, B distribute behind the each patient at random, each time that patient measures the same. Randomized block design require the experiment units are separate in each granule with random granule, can only assign in the granule at random to deal with, the treatment that each experiment unit accepts is different, such as the form 4-9.
It is very similar to repeated measurement design data (form 4-9 ) that measurements and chapter four introduce, such as form 12-3 , and can calculate the variance analytical table (form 12-4 ) that the randomized block design data too. • Table 12-4 The form 12-3 randomized block design data variance analysis table
Table 12-5The pre-post symptom of patient's operation grades (g=2)
Table 4-9 Little white mouse's sarcoma weight after different medicine function（g）
2. It is independent each other of experiment unit in the repeated measurement design, such as form 12-3, i.e. the same experimenter's blood specimen measures are highly relevant, its correlation coefficient is seen the list 12-6. Repeated measurement design like relatively dealing with the difference among the groups by the random granule analysis of variance of chapter four, the precondition is satisfied it is supposed ( examines ) that "sphericity".
Coefficient correlation that each puts the density of blood glucose of time point of the form 12-6 form 12-3 **P<0.01
If the "sphericity" assumptions is met, ANOVA for randomized block design data can be used; If not, ANOVA for randomized block design data can also be used, but it requires to correct the freedom of degree of F value.
§2 repeated measurement design data of two factors and two levels analysis §3 repeated measurement design data of two factors and many levels analysis
One experiment design Treatment---factor A g levels n experiments in each level Time---factor B m time Experiment data: Xijk i=1,2, … ,g j=1,2, … ,m k=1,2, … ,n Experiment data: gmn
Variation and degree of freedom decomposition 1、 *theory：
2、 * theory：
3、 *theory：
Attention: , when reject "sphericity" the degree of freedom must use " sphericity " the coefficient to adjust.
attention • Factorial design: A variance analytical table: The analysis processes the main effect, the correlation. • Repeated measurement design: Two variances analytical tables, processing effect 1, time effect, time and processing correlation 1.
*theory： factorial design： repeated measurement design：
example :12-2According to table 12-2，To treatment group and comparison group, the treatment the diastolic pressure difference carries on the statistical analysis
table 12-13 Comparison between treatment group and comparison group
table 12-12 Around survey comparison and correlation variance analytical table
Attention: Although processing does not have the main effect, it has the correlation with the time, therefore it has the auxiliary effect. 4.Conclusions ?
Example 12-2:According to data 12-2,we have analysis of repeated measurement in three methods at different five time points.
table 12-17 different anesthesia induction, at the same time the patient does not contract press fits estimates the value
variance analytical table according to table 12-14 and the table 12-15 。 • Table 12-18 Different induction method patient systolic pressure comparison variance analytical table
Table 12-19 anesthesia induction and its with induction method correlation variance analytical table
Conclusion: different anesthesia induction method existence group difference (table 12-18), patient's systolic pressure when different induction methods different induction changes tendency different (table 12-19), when A group of different inductions ,the systolic pressure is stable (table 12-20).
Table 12-20 Different anesthesia induction, different time patient's systolic pressure (mmHg）
III Notices 1. It is request that number of objects in each group to be equal.
2. examine "sphericity". 3. repeated measurement design data without a parallel comparison
§4 The situations of the repeated measurement design data statistical analysis commonly misuses.
