QK Box 10.10

Singh et al. (2016) examined the response of copulatory traits in male fruitflies (Drosophila melanogaster) to cold stress. They had two selection treatments (populations selected for cold shock versus control populations, a fixed effect) crossed with five ancestral populations that experimental flies were derived from (random effect). The data we will analyse are from an experiment that compared files from the two selection treatments that were cold shocked, allowed to recover and then copulatory traits were measured after 4, 12 and 30 hours. Different flies were used for each period so there are no repeated measures. This fully-crossed design has two fixed effects (selection treatment, period) and one random effect (ancestral population, termed “block” by Singh et al. (2016), the sample size varied from 35 to 62 flies in each combination of selection treatment, period and ancestral population, and the response variable we will focus on was mating latency (minutes).

The paper is here

Singh, K., Samant, M. A., Tom, M. T. & Prasad, N. G. (2016). Evolution of pre- and post-copulatory traits in male Drosophila melanogaster as a correlated response to selection for resistance to cold stress. PLoS One, 11, e0153629.

Preliminaries

First, load the required packages (afex, car, lattice, lme4, lmerTest, nlme, VCA, Rmisc, MuMin, emmeans)

Import singh data file (singh.csv)

singh <- read.csv("../data/singh.csv")
head(singh,5)

Set contrasts from afex

Convert period and block to factors

set_sum_contrasts()

setting contr.sum globally: options(contrasts=c('contr.sum', 'contr.poly'))

singh$block <- factor(singh$block)
singh$period <- factor(singh$period)

Diagnostics

Check assumptions with boxplots - very skewed distributions

boxplot(matlat~selection*period, data=singh)

Fit OLS anova model to check residuals

singh.aov <- aov(matlat~selection*period*block, data=singh)

Check residuals - variance increases with mean

plot(singh.aov)

Transform response to logs (+1 to handle zero values)

singh$lmatlat <- log10(singh$matlat+1)

Check assumptions - much improved boxplots

boxplot(lmatlat~selection*period, data=singh)

Refit OLS model and check residuals (much better)

singh.aov1 <- aov(lmatlat~selection*period*block, data=singh)
plot(singh.aov1)

Use transformed data

Although the original paper used untransformed data, our pre-analysis checks suggest that a data transformation, or, as in later chapters, a generalized linear model, might be a better fit

Fit lm models to get type III SS using car

singh.lm1 <- lm(lmatlat ~ selection*period*block, data=singh)
summary(singh.lm1)

Call:
lm(formula = lmatlat ~ selection * period * block, data = singh)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.0686 -0.2434 -0.0532  0.1889  1.2209 

Coefficients:
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)                9.16e-01   8.94e-03  102.50  < 2e-16 ***
selection1                 5.83e-02   8.94e-03    6.53  9.3e-11 ***
period1                    1.63e-01   1.27e-02   12.83  < 2e-16 ***
period2                   -1.72e-02   1.26e-02   -1.37  0.17192    
block1                    -7.79e-02   1.89e-02   -4.13  3.8e-05 ***
block2                     6.64e-02   1.82e-02    3.66  0.00026 ***
block3                     1.10e-01   1.74e-02    6.35  3.0e-10 ***
block4                     3.52e-02   1.79e-02    1.97  0.04908 *  
selection1:period1         6.08e-03   1.27e-02    0.48  0.63170    
selection1:period2         6.95e-03   1.26e-02    0.55  0.58136    
selection1:block1         -2.72e-02   1.89e-02   -1.44  0.14888    
selection1:block2          1.77e-02   1.82e-02    0.98  0.32942    
selection1:block3          2.62e-02   1.74e-02    1.50  0.13278    
selection1:block4         -7.89e-03   1.79e-02   -0.44  0.65873    
period1:block1            -9.68e-03   2.67e-02   -0.36  0.71704    
period2:block1             5.37e-02   2.64e-02    2.04  0.04179 *  
period1:block2             5.63e-02   2.61e-02    2.15  0.03133 *  
period2:block2            -5.49e-02   2.56e-02   -2.15  0.03204 *  
period1:block3            -3.02e-02   2.45e-02   -1.23  0.21859    
period2:block3             5.54e-02   2.49e-02    2.23  0.02612 *  
period1:block4             2.98e-02   2.55e-02    1.17  0.24249    
period2:block4            -2.40e-02   2.50e-02   -0.96  0.33585    
selection1:period1:block1 -5.95e-03   2.67e-02   -0.22  0.82391    
selection1:period2:block1  3.47e-02   2.64e-02    1.32  0.18829    
selection1:period1:block2 -4.56e-02   2.61e-02   -1.75  0.08098 .  
selection1:period2:block2 -6.38e-03   2.56e-02   -0.25  0.80311    
selection1:period1:block3 -1.76e-04   2.45e-02   -0.01  0.99427    
selection1:period2:block3 -7.58e-05   2.49e-02    0.00  0.99757    
selection1:period1:block4  4.79e-02   2.55e-02    1.88  0.06015 .  
selection1:period2:block4 -3.49e-03   2.50e-02   -0.14  0.88895    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.341 on 1449 degrees of freedom
Multiple R-squared:  0.208, Adjusted R-squared:  0.192 
F-statistic: 13.1 on 29 and 1449 DF,  p-value: <2e-16

Anova(singh.lm1, type='III')

Anova Table (Type III tests)

Response: lmatlat
                       Sum Sq   Df  F value  Pr(>F)    
(Intercept)              1219    1 10507.20 < 2e-16 ***
selection                   5    1    42.59 9.3e-11 ***
period                     23    2    99.44 < 2e-16 ***
block                      13    4    27.69 < 2e-16 ***
selection:period            0    2     0.53  0.5864    
selection:block             1    4     1.14  0.3338    
period:block                3    8     2.74  0.0054 ** 
selection:period:block      1    8     1.23  0.2755    
Residuals                 168 1449                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Get correct F-ratio and P value for selection x period

f = 0.03/0.14
f

[1] 0.214

1-pf(f, 2, 8, lower.tail = TRUE, log.p = FALSE)

[1] 0.812

Get correct F-ratio and P value for selection

f = 4.72/0.13
f

[1] 36.3

1-pf(f, 1, 4, lower.tail = TRUE, log.p = FALSE)

[1] 0.00382

Get correct F-ratio and P value for period

f = 11.09/0.33
f

[1] 33.6

1-pf(f, 2, 8, lower.tail = TRUE, log.p = FALSE)

[1] 0.000128

Use VCA package to get anova var comps - unreliable sue to imbalance in sample sizes

singh.vca <- anovaMM(lmatlat~selection + period + selection*period + (block) + (selection*block) + (period*block) + (selection*period*block), singh)

Convert variable selection from "character" to "factor"!

VCAinference(singh.vca, alpha=0.05, VarVC=TRUE, excludeNeg=FALSE, constrainCI=FALSE)

Inference from Mixed Model Fit
------------------------------

> VCA Result:
-------------

    [Fixed Effects]

                  int          selectionFCB          selectionFSB               period4              period12              period30  selectionFCB:period4  selectionFSB:period4 
               0.7235                0.0967                0.0000                0.2906                0.1100                0.0000                0.0332                0.0000 
selectionFCB:period12 selectionFSB:period12 selectionFCB:period30 selectionFSB:period30 
               0.0322                0.0000                0.0000                0.0000 


    [Variance Components]

  Name                   DF       SS       MS     VC     %Total  SD     CV[%]   Var(VC)
1 total                  419.9978                 0.1282 100     0.358  39.263         
2 block                  4        12.7393  3.1848 0.0097 7.5916  0.0986 10.8181 1e-04  
3 selection:block        4        0.523    0.1307 0*     0*      0*     0*      0      
4 period:block           8        2.6243   0.328  0.0019 1.4688  0.0434 4.7585  0      
5 selection:period:block 8        1.1446   0.1431 6e-04  0.4328  0.0236 2.5829  0      
6 error                  1449     168.1046 0.116  0.116  90.5068 0.3406 37.3529 0      

Mean: 0.912 (N = 1479) 

Experimental Design: unbalanced  |  Method: ANOVA | * VC set to 0 | adapted MS used for total DF


> VC:
-----
                       Estimate CI LCL CI UCL One-Sided LCL One-Sided UCL
total                  0.1282   0.1125 0.1475 0.1148        0.1441       
block                  0.0097   0*     0.0248 0*            0.0224       
selection:block        0        0*     0.0016 0*            0.0013       
period:block           0.0019   0*     0.0055 0*            0.0049       
selection:period:block 6e-04    0*     0.0034 0*            0.003        
error                  0.116    0.108  0.1249 0.1093        0.1235       

> SD:
-----
                       Estimate CI LCL CI UCL One-Sided LCL One-Sided UCL
total                  0.358    0.3354 0.384  0.3389        0.3797       
block                  0.0986   0*     0.1575 0*            0.1496       
selection:block        0        0*     0.0395 0*            0.0362       
period:block           0.0434   0*     0.074  0*            0.0699       
selection:period:block 0.0236   0*     0.0586 0*            0.0545       
error                  0.3406   0.3286 0.3535 0.3305        0.3514       

> CV[%]:
--------
                       Estimate CI LCL  CI UCL  One-Sided LCL One-Sided UCL
total                  39.263   36.778  42.111  37.164        41.6359      
block                  10.8181  0*      17.2708 0*            16.4055      
selection:block        0        0*      4.3363  0*            3.9724       
period:block           4.7585   0*      8.1098  0*            7.6705       
selection:period:block 2.5829   0*      6.4271  0*            5.9782       
error                  37.3529  36.0412 38.7644 36.2481       38.533       


95% Confidence Level  | * CI-limits constrained to be >= 0
SAS PROC MIXED method used for computing CIs 

Get least squares means for selection and period

emmeans (singh.lm1, "selection")

NOTE: Results may be misleading due to involvement in interactions

 selection emmean     SE   df lower.CL upper.CL
 FCB        0.974 0.0130 1449    0.949    1.000
 FSB        0.858 0.0123 1449    0.834    0.882

Results are averaged over the levels of: period, block 
Confidence level used: 0.95 

emmeans(singh.lm1, "period")

NOTE: Results may be misleading due to involvement in interactions

 period emmean     SE   df lower.CL upper.CL
 4       1.079 0.0156 1449    1.048    1.109
 12      0.899 0.0154 1449    0.869    0.929
 30      0.770 0.0155 1449    0.740    0.801

Results are averaged over the levels of: selection, block 
Confidence level used: 0.95 

Backtransform values

FCB & FSB:

#means
10^.974

[1] 9.42

10^.858

[1] 7.21

#lower CIs
10^.949

[1] 8.89

10^ .834

[1] 6.82

#upper CIs
10^1

[1] 10

10^.882

[1] 7.62

Periods

10^1.079

[1] 12

10^.899

[1] 7.93

10^.77

[1] 5.89

Fit mixed effects model using REML

singh.lmer1 <- lmer(lmatlat~selection + period +selection:period + (1|block) + (1|selection:block)
            + (1|period:block) + (1|selection:period:block), REML=TRUE, singh)

boundary (singular) fit: see help('isSingular')

summary(singh.lmer1, ddf="Kenward-Roger")

Linear mixed model fit by REML. t-tests use Kenward-Roger's method ['lmerModLmerTest']
Formula: lmatlat ~ selection + period + selection:period + (1 | block) +      (1 | selection:block) + (1 | period:block) + (1 | selection:period:block)
   Data: singh

REML criterion at convergence: 1072

Scaled residuals: 
   Min     1Q Median     3Q    Max 
-3.219 -0.706 -0.118  0.558  3.666 

Random effects:
 Groups                 Name        Variance Std.Dev.
 selection:period:block (Intercept) 0.000473 0.0217  
 period:block           (Intercept) 0.001876 0.0433  
 selection:block        (Intercept) 0.000000 0.0000  
 block                  (Intercept) 0.009328 0.0966  
 Residual                           0.116030 0.3406  
Number of obs: 1479, groups:  selection:period:block, 30; period:block, 15; selection:block, 10; block, 5

Fixed effects:
                   Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)         0.91634    0.04568  4.00065   20.06  3.6e-05 ***
selection1          0.05922    0.00977  3.95933    6.06   0.0039 ** 
period1             0.16275    0.02103  8.02693    7.74  5.4e-05 ***
period2            -0.01835    0.02098  7.97861   -0.87   0.4072    
selection1:period1  0.00571    0.01384  7.97814    0.41   0.6909    
selection1:period2  0.00519    0.01377  7.90483    0.38   0.7160    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) slctn1 perid1 perid2 slc1:1
selection1   0.010                            
period1      0.001  0.005                     
period2     -0.001  0.011 -0.500              
slctn1:prd1  0.001  0.007  0.033 -0.026       
slctn1:prd2  0.004 -0.004 -0.026  0.038 -0.501
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

anova(singh.lmer1, ddf="Kenward-Roger")

Type III Analysis of Variance Table with Kenward-Roger's method
                 Sum Sq Mean Sq NumDF DenDF F value Pr(>F)    
selection          4.26    4.26     1  3.96   36.73 0.0039 ** 
period             8.34    4.17     2  8.00   35.96 0.0001 ***
selection:period   0.07    0.04     2  7.92    0.31 0.7400    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Model fit singular - over-fitted due to zero variance component for 3 way interaction

CI on REML variance components (remembering to square CIs from lmer with are in SD units)

singh.ci1 <- confint.merMod(singh.lmer1, oldNames=FALSE)

Computing profile confidence intervals ...
Warning: slightly lower deviances (diff=-1.13687e-12) detectedWarning: Last two rows have identical or NA .zeta values: using minstepWarning: Last two rows have identical or NA .zeta values: using minstepWarning: non-monotonic profile for sd_(Intercept)|selection:period:blockWarning: unexpected decrease in profile: using minstepWarning: unexpected decrease in profile: using minstepWarning: unexpected decrease in profile: using minstepWarning: unexpected decrease in profile: using minstepWarning: non-monotonic profile for selection1Warning: bad spline fit for sd_(Intercept)|selection:period:block: falling back to linear interpolationWarning: collapsing to unique 'x' valuesWarning: bad spline fit for selection1: falling back to linear interpolation

singh.vc1 <- (singh.ci1)^2
print(singh.vc1)

                                         2.5 %   97.5 %
sd_(Intercept)|selection:period:block 0.00e+00 0.002792
sd_(Intercept)|period:block           5.12e-05 0.006111
sd_(Intercept)|selection:block        0.00e+00 0.002572
sd_(Intercept)|block                  2.10e-03 0.038632
sigma                                 1.08e-01 0.124796
(Intercept)                           6.70e-01 1.029152
selection1                            1.50e-03 0.006291
period1                               1.50e-02 0.041501
period2                               3.49e-03 0.000438
selection1:period1                    3.65e-04 0.000920
selection1:period2                    3.89e-04 0.000869

Refit model omitting three way interaction to deal with singular fit

singh.lmer2 <- lmer(lmatlat~selection + period +selection:period + (1|block) + (1|selection:block)
                     + (1|period:block), REML=TRUE, singh)

Compare models

anova(singh.lmer1, singh.lmer2)

refitting model(s) with ML (instead of REML)

Data: singh
Models:
singh.lmer2: lmatlat ~ selection + period + selection:period + (1 | block) + (1 | selection:block) + (1 | period:block)
singh.lmer1: lmatlat ~ selection + period + selection:period + (1 | block) + (1 | selection:block) + (1 | period:block) + (1 | selection:period:block)
            npar  AIC  BIC logLik deviance Chisq Df Pr(>Chisq)
singh.lmer2   10 1054 1107   -517     1034                    
singh.lmer1   11 1056 1115   -517     1034     0  1          1

AICc(singh.lmer1, singh.lmer2)

Omitting the interaction has little effect on fit

Get results from this reduced model

summary(singh.lmer2, ddf="Kenward-Roger")

Linear mixed model fit by REML. t-tests use Kenward-Roger's method ['lmerModLmerTest']
Formula: lmatlat ~ selection + period + selection:period + (1 | block) +      (1 | selection:block) + (1 | period:block)
   Data: singh

REML criterion at convergence: 1073

Scaled residuals: 
   Min     1Q Median     3Q    Max 
-3.222 -0.705 -0.121  0.553  3.670 

Random effects:
 Groups          Name        Variance Std.Dev.
 period:block    (Intercept) 2.12e-03 0.04610 
 selection:block (Intercept) 8.77e-05 0.00937 
 block           (Intercept) 9.29e-03 0.09640 
 Residual                    1.16e-01 0.34085 
Number of obs: 1479, groups:  period:block, 15; selection:block, 10; block, 5

Fixed effects:
                    Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)         9.16e-01   4.57e-02  4.00e+00   20.05  3.6e-05 ***
selection1          5.93e-02   9.41e-03  3.95e+00    6.31   0.0034 ** 
period1             1.63e-01   2.11e-02  8.03e+00    7.74  5.4e-05 ***
period2            -1.85e-02   2.10e-02  7.98e+00   -0.88   0.4035    
selection1:period1  5.78e-03   1.26e-02  1.46e+03    0.46   0.6469    
selection1:period2  4.89e-03   1.26e-02  1.46e+03    0.39   0.6972    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) slctn1 perid1 perid2 slc1:1
selection1   0.010                            
period1      0.001  0.005                     
period2     -0.001  0.011 -0.500              
slctn1:prd1  0.002  0.008  0.036 -0.028       
slctn1:prd2  0.004 -0.004 -0.028  0.041 -0.502

F tests of fixed effects (Type III SS) for this new model - same conclusions as above

anova(singh.lmer2, ddf="Kenward-Roger")

Type III Analysis of Variance Table with Kenward-Roger's method
                 Sum Sq Mean Sq NumDF DenDF F value Pr(>F)    
selection          4.62    4.62     1     4   39.80 0.0034 ** 
period             8.34    4.17     2     8   35.89 0.0001 ***
selection:period   0.08    0.04     2  1462    0.36 0.6969    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Calculate CIs on var component for this new model

singh.ci2 <- confint.merMod(singh.lmer2, oldNames=FALSE)

Computing profile confidence intervals ...
Warning: slightly lower deviances (diff=-1.36424e-12) detectedWarning: unexpected decrease in profile: using minstepWarning: Last two rows have identical or NA .zeta values: using minstepWarning: non-monotonic profile for sd_(Intercept)|selection:blockWarning: bad spline fit for sd_(Intercept)|selection:block: falling back to linear interpolationWarning: collapsing to unique 'x' values

singh.vc2 <- (singh.ci2)^2
print(singh.vc2)

                                  2.5 %   97.5 %
sd_(Intercept)|period:block    4.89e-05 0.006111
sd_(Intercept)|selection:block 0.00e+00 0.002659
sd_(Intercept)|block           2.07e-03 0.038632
sigma                          1.08e-01 0.124796
(Intercept)                    6.71e-01 1.027682
selection1                     1.76e-03 0.005897
period1                        1.57e-02 0.040132
period2                        3.14e-03 0.000356
selection1:period1             3.64e-04 0.000920
selection1:period2             3.89e-04 0.000869

Analysis using untransformed response

This is the approach used by the paper’s author’s. We’ve given the initial OLS fit, which you can adjust for the random effects by recalculating F-ratios as shown in the earlier code. You can easily also run the REML mixed effects model.

options(digits=4)
singh.lm2 <- lm(matlat~selection*period*block, data=singh)
summary(singh.lm2)

Call:
lm(formula = matlat ~ selection * period * block, data = singh)

Residuals:
   Min     1Q Median     3Q    Max 
-22.80  -6.94  -3.38   1.22 100.30 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                11.9362     0.3983   29.97  < 2e-16 ***
selection1                  2.0177     0.3983    5.07  4.6e-07 ***
period1                     5.6324     0.5656    9.96  < 2e-16 ***
period2                    -1.0621     0.5615   -1.89   0.0587 .  
block1                     -2.1440     0.8405   -2.55   0.0108 *  
block2                      1.8845     0.8090    2.33   0.0200 *  
block3                      3.2913     0.7750    4.25  2.3e-05 ***
block4                      1.1268     0.7957    1.42   0.1569    
selection1:period1          0.8168     0.5656    1.44   0.1489    
selection1:period2          0.1700     0.5615    0.30   0.7621    
selection1:block1          -1.3873     0.8405   -1.65   0.0990 .  
selection1:block2          -0.1761     0.8090   -0.22   0.8277    
selection1:block3           1.4173     0.7750    1.83   0.0676 .  
selection1:block4           0.5771     0.7957    0.73   0.4684    
period1:block1             -1.4099     1.1906   -1.18   0.2366    
period2:block1              1.6193     1.1747    1.38   0.1683    
period1:block2              3.4923     1.1639    3.00   0.0027 ** 
period2:block2             -2.4888     1.1397   -2.18   0.0291 *  
period1:block3             -0.3059     1.0925   -0.28   0.7795    
period2:block3              1.7296     1.1083    1.56   0.1188    
period1:block4              0.2358     1.1360    0.21   0.8356    
period2:block4              0.0091     1.1132    0.01   0.9935    
selection1:period1:block1  -0.8870     1.1906   -0.75   0.4564    
selection1:period2:block1   1.5682     1.1747    1.33   0.1821    
selection1:period1:block2  -0.8039     1.1639   -0.69   0.4898    
selection1:period2:block2  -0.6862     1.1397   -0.60   0.5472    
selection1:period1:block3  -0.1096     1.0925   -0.10   0.9201    
selection1:period2:block3   0.0743     1.1083    0.07   0.9465    
selection1:period1:block4   1.7904     1.1360    1.58   0.1152    
selection1:period2:block4  -0.0457     1.1132   -0.04   0.9673    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.2 on 1449 degrees of freedom
Multiple R-squared:  0.13,  Adjusted R-squared:  0.113 
F-statistic: 7.48 on 29 and 1449 DF,  p-value: <2e-16

Anova(singh.lm2, type='III')

Anova Table (Type III tests)

Response: matlat
                       Sum Sq   Df F value  Pr(>F)    
(Intercept)            206972    1  898.08 < 2e-16 ***
selection                5914    1   25.66 4.6e-07 ***
period                  25814    2   56.01 < 2e-16 ***
block                   11609    4   12.59 4.5e-10 ***
selection:period          805    2    1.75   0.175    
selection:block          1278    4    1.39   0.236    
period:block             4086    8    2.22   0.024 *  
selection:period:block   1411    8    0.77   0.634    
Residuals              333938 1449                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#use anova to show MS
anova(singh.lm2)

Analysis of Variance Table

Response: matlat
                         Df Sum Sq Mean Sq F value  Pr(>F)    
selection                 1   5766    5766   25.02 6.4e-07 ***
period                    2  25108   12554   54.47 < 2e-16 ***
block                     4  11607    2902   12.59 4.5e-10 ***
selection:period          2    637     319    1.38   0.251    
selection:block           4   1255     314    1.36   0.245    
period:block              8   4178     522    2.27   0.021 *  
selection:period:block    8   1411     176    0.77   0.634    
Residuals              1449 333938     230                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

glance(singh.lm2)