1. Analysis of Variance (ANOVA)
9/2/2019
Analysis of Variance (ANOVA)
SPH 247
Statistical Analysis of
Laboratory Data
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

2. ANOVA—Fixed and Random Effects
We will review the analysis of variance (ANOVA) and then move to random and fixed effects models
Nested models are used to look at levels of variability (days within subjects, replicate measurements within days)
Crossed models are often used when there are both fixed and random effects.
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

3. 9/2/2019
The Basic Idea
The analysis of variance is a way of testing whether observed differences between groups are too large to be explained by chance variation
One-way ANOVA is used when there are k ≥ 2 groups for one factor, and no other quantitative variable or classification factor.
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

4. 9/2/2019 A B C 9 10 12 7 14 8
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SPH 247 Statistical Analysis of Laboratory Data

5. Column Deviations from grand mean + Cell Deviations from column mean
9/2/2019
Data = Grand Mean +
Column Deviations from grand mean +
Cell Deviations from column mean
Are the column deviations from the grand mean
too big to be accounted for by the cell
deviations from the column means?
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

6. 9/2/2019
Data A B C 9 10 12 7 14 8
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SPH 247 Statistical Analysis of Laboratory Data

7. Column Means A B C 8 9 13 9/2/2019 March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

8. Deviations from Column Means
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Deviations from Column Means A B C 1 -1
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SPH 247 Statistical Analysis of Laboratory Data

9. red.cell.folate package:ISwR R Documentation Red cell folate data
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red.cell.folate package:ISwR R Documentation
Red cell folate data
Description:
The 'folate' data frame has 22 rows and 2 columns. It contains
data on red cell folate levels in patients receiving three
different methods of ventilation during anesthesia.
Format:
This data frame contains the following columns:
folate: a numeric vector. Folate concentration (μg/l).
ventilation: a factor with levels:
'N2O+O2,24h': 50% nitrous oxide and 50% oxygen, continuously for 24~hours;
'N2O+O2,op': 50% nitrous oxide and 50% oxygen, only during operation;
'O2,24h': no nitrous oxide, but 35-50% oxygen for 24~hours.
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

10. > data(red.cell.folate) > help(red.cell.folate)
9/2/2019
> library(ISwR)
> data(red.cell.folate)
> help(red.cell.folate)
> summary(red.cell.folate)
folate ventilation
Min. : N2O+O2,24h:8
1st Qu.: N2O+O2,op :9
Median : O2,24h :5
Mean :283.2
3rd Qu.:305.5
Max. : > attach(red.cell.folate) > plot(folate ~ ventilation)
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

11. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

12. > folate.lm <- lm(folate ~ ventilation) > summary(folate.lm)
9/2/2019
> folate.lm <- lm(folate ~ ventilation)
> summary(folate.lm)
Call:
lm(formula = folate ~ ventilation)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-14 ***
ventilationN2O+O2,op *
ventilationO2,24h
---
Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: on 19 degrees of freedom
Multiple R-Squared: , Adjusted R-squared:
F-statistic: on 2 and 19 DF, p-value:
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

13. Analysis of Variance Table Response: folate
9/2/2019
> anova(folate.lm)
Analysis of Variance Table
Response: folate
Df Sum Sq Mean Sq F value Pr(>F)
ventilation *
Residuals
---
Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> TukeyHSD(aov(folate~ventilation))
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = folate ~ ventilation)
$ventilation
diff lwr upr p adj
N2O+O2,op-N2O+O2,24h
O2,24h-N2O+O2,24h
O2,24h-N2O+O2,op
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

14. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

15. Two- and Multi-way ANOVA
9/2/2019
Two- and Multi-way ANOVA
If there is more than one factor, the sum of squares can be decomposed according to each factor, and possibly according to interactions
One can also have factors and quantitative variables in the same model (cf. analysis of covariance)
All have similar interpretations
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

16. Heart rates after enalaprilat (ACE inhibitor) Description:
9/2/2019
Heart rates after enalaprilat (ACE inhibitor)
Description:
36 rows and 3 columns.
data for nine patients with congestive heart failure before and
shortly after administration of enalaprilat, in a balanced two-way
layout.
Format:
hr a numeric vector. Heart rate in beats per minute.
subj a factor with levels '1' to '9'.
time a factor with levels '0' (before), '30', '60', and '120'
(minutes after administration).
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

17. > attach(heart.rate) > heart.rate hr subj time 1 96 1 0
9/2/2019
> data(heart.rate)
> attach(heart.rate)
> heart.rate
hr subj time
......
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

18. > hr.lm <- lm(hr~subj+time) > anova(hr.lm)
9/2/2019
> plot(hr~subj)
> plot(hr~time)
> hr.lm <- lm(hr~subj+time)
> anova(hr.lm)
Analysis of Variance Table
Response: hr
Df Sum Sq Mean Sq F value Pr(>F)
subj e-16 ***
time *
Residuals
---
Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> sres <- resid(lm(hr~subj))
> plot(sres~time)
Note that when the design is orthogonal, the ANOVA results don’t depend on the order of terms.
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SPH 247 Statistical Analysis of Laboratory Data

19. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

20. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

21. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

22. Fixed and Random Effects
A fixed effect is a factor that can be duplicated (dosage of a drug)
A random effect is one that cannot be duplicated
Patient/subject
Repeated measurement
There can be important differences in the analysis of data with random effects
The error term is always a random effect
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

23. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

24. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

25. Estradiol data from Rosner
5 subjects from the Nurses’ Health Study
One blood sample each
Each sample assayed twice for estradiol (and three other hormones)
The within-subject variability is strictly technical/assay
Variability within a person over time will be much greater
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

26. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

27. > anova(lm(Estradiol ~ Subject,data=endocrin))
Analysis of Variance Table
Response: Estradiol
Df Sum Sq Mean Sq F value Pr(>F)
Subject **
Residuals
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Replication error variance is 6.043, so the standard deviation of replicates is 2.46 pg/mL
This compared to average levels across subjects from 8.05 to 18.80
Estimated variance across subjects is ( − 6.043)/2 =
Standard deviation across subjects is 8.43 pg/mL
If we average the replicates, we get five values, the standard deviation of which is also 8.43
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

28. Fasting Blood Glucose
Part of a larger study that also examined glucose tolerance during pregnancy
Here we have 53 subjects with 6 tests each at intervals of at least a year
The response is glucose as mg/100mL
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

29. > anova(lm(FG ~ Subject,data=fg2)) Analysis of Variance Table
Response: FG
Df Sum Sq Mean Sq F value Pr(>F)
Subject e-09 ***
Residuals
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >
Estimated within-Subject variance is , so the standard deviation is 8.48 mg/100mL
Estimated between-Subject variance is ( − )/6 = , sd = 4.80 mg/100mL
The variance of the 53 means is 35.05, which is larger because it includes a component of the
within-subject variance
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

30. Nested Random Effects Models
Cooperative trial with 6 laboratories, one analyte (7 in the full data set), 3 batches per lab (a month apart), and 2 replicates per batch
Estimate the variance components due to labs, batches, and replicates
Test for significance if possible
Effects are lab, batch-in-lab, and error
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

31. > coop2 <- coop[coop$Spc=="S1",] > summary(coop2)
> library(MASS)
> data(coop)
> names(coop)
[1] "Lab" "Spc" "Bat" "Conc"
> summary(coop)
Lab Spc Bat Conc
L1:42 S1:36 B1:84 Min. :0.1100
L2:42 S2:36 B2:84 1st Qu.:0.4675
L3:42 S3:36 B3:84 Median :1.0600
L4:42 S4: Mean :1.9215
L5:42 S5: rd Qu.:1.7000
L6:42 S6: Max. :9.9000
S7:36
> coop2 <- coop[coop$Spc=="S1",]
> summary(coop2)
Lab Spc Bat Conc
L1:6 S1:36 B1:12 Min. :0.2900
L2:6 S2: 0 B2:12 1st Qu.:0.3575
L3:6 S3: 0 B3:12 Median :0.4000
L4:6 S4: Mean :0.5081
L5:6 S5: rd Qu.:0.4600
L6:6 S6: Max. :1.3000
S7: 0
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SPH 247 Statistical Analysis of Laboratory Data

32. March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

33. Analysis using lm or aov
> anova(lm(Conc ~ Lab + Lab:Bat,data=coop2))
Analysis of Variance Table
Response: Conc
Df Sum Sq Mean Sq F value Pr(>F)
Lab e-10 ***
Lab:Bat *
Residuals
The test for batch-in-lab is correct, but the test for lab is not—the denominator should be
The Lab:Bat MS, so F(5,12) = / = and p = 3.47e-4, still significant
Residual
Batch
Lab
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SPH 247 Statistical Analysis of Laboratory Data

34. Analysis using lmer R package lme4
One term per effect, in this case nested
In this case, no fixed effects
> lmer(Conc ~ 1+(1|Lab)+(1|Bat:Lab),data=coop2)
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SPH 247 Statistical Analysis of Laboratory Data

35. > lmer(Conc ~ 1+(1|Lab)+(1|Bat:Lab),data=coop2)
> library(lme4)
> lmer(Conc ~ 1+(1|Lab)+(1|Bat:Lab),data=coop2)
Linear mixed model fit by REML ['lmerMod']
Formula: Conc ~ 1 + (1 | Lab) + (1 | Bat:Lab)
Data: coop2
REML criterion at convergence:
Random effects:
Groups Name Std.Dev.
Bat:Lab (Intercept)
Lab (Intercept)
Residual
Number of obs: 36, groups: Bat:Lab, 18; Lab, 6
Fixed Effects:
(Intercept)
0.5081
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

36. Hypothesis Tests
When data are balanced, one can compute expected mean squares, and many times can compute a valid F test.
In more complex cases, or when data are unbalanced, this is more difficult
One requirement for certain hypothesis tests to be valid is that the null hypothesis value is not on the edge of the possible values
For H0: α = 0, we have that α could be either positive or negative
For H0: σ2 = 0, negative variances are not possible
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

37. Effect Variance SD
Residual
Batch
Lab
The variance among replicates a month apart ( = ) is about twice that of those on the same day ( ), and the standard deviations are and These are CV’s on the average of 21% and 16% respectively
The variance among values from different labs is about = , with a standard deviation of and a CV of about 52%
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data

38. More complex models
When data are balanced and the expected mean squares can be computed, this is a valid way for testing and estimation
Programs like lme and lmer in R and Proc Mixed in SAS can handle complex models
But most likely this is a time when you may need to consult an expert
March 31, 2015
SPH 247 Statistical Analysis of Laboratory Data