SPH 247 Statistical Analysis of Laboratory Data

Transcription

1 SPH 247 Statistical Analysis of Laboratory Data March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 1

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 2

3 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 3

4 A B C March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 4

5 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 5

6 Data A B C March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 6

7 Column Means A B C March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 7

8 Deviations from Column Means A B C March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 8

9 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 9

10 > library(iswr) > data(red.cell.folate) > help(red.cell.folate) > summary(red.cell.folate) folate ventilation Min. :206.0 N2O+O2,24h:8 1st Qu.:249.5 N2O+O2,op :9 Median :274.0 O2,24h :5 Mean : rd Qu.:305.5 Max. :392.0 > attach(red.cell.folate) > plot(folate ~ ventilation) March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 10

11 folate N2O+O2,24h N2O+O2,op O2,24h ventilation March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 11

12 > folate.lm <- lm(folate ~ ventilation) > summary(folate.lm) Call: lm(formula = folate ~ ventilation) Residuals: Min 1Q Median 3Q 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 12

13 > 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 13

14 folate N2O+O2,24h N2O+O2,op O2,24h ventilation March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 14

15 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 15

16 Heart rates after enalaprilat (ACE inhibitor) Description: Format: 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. 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 16

17 > data(heart.rate) > attach(heart.rate) > heart.rate hr subj time March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 17

18 > plot(hr~subj) > plot(hr~time) > hr.lm <- lm(hr~subj+time) > anova(hr.lm) Analysis of Variance Table Note that when the design is orthogonal, the ANOVA results don t depend on the order of terms. 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) March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 18

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

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

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

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 22

23 Fixed Effect One-way ANOVA y y α = µ µ i = µ + 2 ~ (0, ) 0 α = 0 E( MSE) i N i = µ + α + i 2 EM ( SA) = Q( α ) + σ n α Q( αi ) = a 1 H : Q( α ) = 0 i i i σ = σ i i 2 i MSA / MSE ~ F under the null 2 March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 23

24 Random Effect One-way ANOVA y y α 2 ~ (0, σ ) 2 ~ (0, σα ) E( MSE) 0 N i N i i 2 : σα = 0 2 E( MSA) = nσ 2 2 α n is replicates per level of α H = σ MSA / MSE ~ F under the null 2 ˆ α = ( )/ σ = µ + = µ + α + i + σ MSA MSE n March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 24

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 25

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

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 * 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 Estimated variance across subjects is ( )/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 27

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 28

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 * > 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 29

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 30

31 > library(mass) > data(coop) > names(coop) [1] "Lab" "Spc" "Bat" "Conc" > summary(coop) Lab Spc Bat Conc L1:42 S1:36 B1:84 Min. : L2:42 S2:36 B2:84 1st Qu.: L3:42 S3:36 B3:84 Median : L4:42 S4:36 Mean : L5:42 S5:36 3rd Qu.: L6:42 S6:36 Max. : S7:36 > coop2 <- coop[coop$spc=="s1",] > summary(coop2) Lab Spc Bat Conc L1:6 S1:36 B1:12 Min. : L2:6 S2: 0 B2:12 1st Qu.: L3:6 S3: 0 B3:12 Median : L4:6 S4: 0 Mean : L5:6 S5: 0 3rd Qu.: L6:6 S6: 0 Max. : S7: 0 March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 31

32 Expected Mean Squares l laboratories b batches per laboratory r replicates per batch laboratories E( MS ) = brσ + rσ + σ 2 ˆ B = ( L L B e batches within laboratories E( MS ) 2 2 B B e replicates within batches E( MSE) σ MS = rσ + σ = σ B 2 e MSE)/ r 2 ˆ L = ( L B )/ σ MS MS br 2 Equal under the null σ L = 0 March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 32

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 March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 33

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) March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 34

35 > 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) March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 35

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 H 0 : α = 0, we have that α could be either positive or negative For H 0 : σ 2 = 0, negative variances are not possible March 31, 2015 SPH 247 Statistical Analysis of Laboratory Data 36

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 37

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 38