Modified from hybridViability.Rmd (Oct 2017)

Guides

Purpose

Identify reproductive barriers between two sympatric moth-pollinated plant species, Schiedea kaalae and S. hookeri by fitting a generalized linear mixed model (GLMM).

In the experimental design, the following crosstypes were made:

  • within species, between population (may show outbreeding depression or heterosis)
  • within species, within populations (may show inbreeding depression)
  • hybrids between species (indicates species barrier from pollination to seed production)

In this analysis the response variable is the pollen viability of each cross. Other barriers (hybrid survival, flowering) could be analyzed in a similar framework, with appropriate changes to the underlying distribution.

Fixed effects:

  • crosstype - hybrids, within population, between populations
  • species - species of the maternal plant that produced the Viability

Potential random effects:

  • mompop - maternal plant population
  • mompid - maternal plant, specified by its population and ID
  • dadpop - paternal plant population

Data Import

pol <- read.table("hybridpollen.csv", header=T, sep="\t", colClasses=c(date="Date"))

pol$v <- rowSums(pol[,grepl("V", colnames(pol))])/10 #average viable counts
pol$i <- rowSums(pol[,grepl("I", colnames(pol))])/10 #average inviable counts
pol$vpf <- (pol$v * 444.4/20) * 10      #number viable per flower (5 flrs)
pol$ipf <- (pol$i * 444.4/20) * 10      #number inviable per flower (5 flrs)
pol$vp <- pol$vpf / (pol$vpf + pol$ipf) #proportion viable
pol$tpf <- pol$vpf + pol$ipf

pol <- pol[pol$crosstype!="field",]

momdadid <- as.data.frame(matrix(unlist(matrix(lapply(matrix(unlist(strsplit(as.character(pol$fullcross), " x ", fixed=T)), nrow=2), function(x) { spl <- unlist(strsplit(unlist(strsplit(x, " ", fixed=T)), "-", fixed=T)); c(spl[1],gsub("-NA","",paste(spl[2],spl[3],spl[4], sep="-"))) }), ncol=2)), ncol=4, byrow=T))
colnames(momdadid) <- c("mompop","momid","dadpop","dadid")

pol <- cbind(pol, momdadid)
pol$species
   NULL
#treat populations as factors
pol$species <- factor(ifelse(pol$mompop %in% c("904","3587", "892"), "kaal", "hook"))
pol$dadsp <-   factor(ifelse(pol$dadpop %in% c("904","3587", "892"), "kaal", "hook"))
pol$cross <-   factor(toupper(paste0(substr(pol$species,0,1), substr(pol$dadsp,0,1))))
crosscol <- c("green","blue","orange","red")

#rename crosstype codes
pol$crosstype <- factor(pol$crosstype, levels=c("between", "within", "hybrid"))
#made "between" the first reference level to facilitate comparison between outcrossing populations and hybridizing species 

pol$mompop <- sapply(pol$mompop, mapvalues, from = c("794","866","899","879","892","904","3587"), to = c("WK","WK","WK","879WKG","892WKG","904WPG","3587WP"))
pol$dadpop <- sapply(pol$dadpop, mapvalues, from = c("794","866","899","879","892","904","3587"), to = c("WK","WK","WK","879WKG","892WKG","904WPG","3587WP"), warn_missing=F)

#define interactions
pol <- within(pol, sxc <- interaction(species,crosstype))
pol <- within(pol, sxcxm <- interaction(species,crosstype,mompop,momid))
pol <- within(pol, mompid <- as.factor(paste(mompop,momid,sep=".")))
pol <- within(pol, dadpid <- as.factor(paste(dadpop,dadid,sep=".")))
pol <- within(pol, smompop <- as.factor(paste(species,mompop,sep="")))
#check final structure
str(pol)
   'data.frame':    197 obs. of  45 variables:
    $ date     : Date, format: "15-10-19" "16-08-17" ...
    $ cell     : int  3 8 2 5 3 1 3 3 4 NA ...
    $ fullcross: Factor w/ 105 levels "3587-12","3587-14",..: 54 54 54 53 53 53 53 53 53 53 ...
    $ V1       : int  30 17 13 16 20 3 26 26 24 5 ...
    $ I1       : int  28 54 40 43 36 42 35 43 62 42 ...
    $ V2       : int  24 2 20 18 11 6 21 50 18 6 ...
    $ I2       : int  42 29 27 46 35 31 35 37 71 68 ...
    $ V3       : int  26 9 5 17 12 0 24 24 12 4 ...
    $ I3       : int  33 43 23 57 34 33 21 27 59 36 ...
    $ V4       : int  30 10 6 21 24 6 30 49 15 5 ...
    $ I4       : int  28 50 37 36 47 23 44 37 53 55 ...
    $ V5       : int  24 10 12 13 10 4 19 40 9 5 ...
    $ I5       : int  37 41 36 57 32 35 30 32 27 48 ...
    $ V6       : int  26 14 7 13 12 5 30 41 19 5 ...
    $ I6       : int  35 43 49 39 21 26 28 28 47 50 ...
    $ V7       : int  37 7 7 22 13 1 31 31 15 2 ...
    $ I7       : int  45 54 42 55 37 33 25 31 43 60 ...
    $ V8       : int  40 14 8 15 8 2 15 30 24 6 ...
    $ I8       : int  43 59 44 47 27 29 46 30 40 42 ...
    $ V9       : int  38 14 8 23 26 1 24 44 7 4 ...
    $ I9       : int  25 50 37 65 43 32 42 57 53 69 ...
    $ V10      : int  33 12 14 14 12 2 33 50 14 7 ...
    $ I10      : int  57 54 44 37 39 33 37 49 47 59 ...
    $ crosstype: Factor w/ 3 levels "between","within",..: 3 3 3 3 3 3 3 3 3 3 ...
    $ set      : int  2 2 2 2 2 2 2 2 2 NA ...
    $ id       : Factor w/ 131 levels "","1","10","102-7",..: 93 112 19 67 93 104 3 3 44 68 ...
    $ notes    : Factor w/ 8 levels "","3587-14 (4/3/13)",..: 1 1 1 1 1 1 1 1 1 1 ...
    $ v        : num  30.8 10.9 10 17.2 14.8 3 25.3 38.5 15.7 4.9 ...
    $ i        : num  37.3 47.7 37.9 48.2 35.1 31.7 34.3 37.1 50.2 52.9 ...
    $ vpf      : num  6844 2422 2222 3822 3289 ...
    $ ipf      : num  8288 10599 8421 10710 7799 ...
    $ vp       : num  0.452 0.186 0.209 0.263 0.297 ...
    $ tpf      : num  15132 13021 10643 14532 11088 ...
    $ mompop   : Factor w/ 5 levels "3587WP","WK",..: 3 3 3 3 3 3 3 3 3 3 ...
    $ momid    : Factor w/ 16 levels "1","10","10-1",..: 7 7 7 7 7 7 7 7 7 7 ...
    $ dadpop   : Factor w/ 5 levels "3587WP","WK",..: 4 4 4 4 4 4 4 4 4 4 ...
    $ dadid    : Factor w/ 14 levels "1","10","10-1",..: 9 9 9 2 2 2 2 2 2 2 ...
    $ species  : Factor w/ 2 levels "hook","kaal": 1 1 1 1 1 1 1 1 1 1 ...
    $ dadsp    : Factor w/ 2 levels "hook","kaal": 2 2 2 2 2 2 2 2 2 2 ...
    $ cross    : Factor w/ 4 levels "HH","HK","KH",..: 2 2 2 2 2 2 2 2 2 2 ...
    $ sxc      : Factor w/ 6 levels "hook.between",..: 5 5 5 5 5 5 5 5 5 5 ...
    $ sxcxm    : Factor w/ 480 levels "hook.between.3587WP.1",..: 197 197 197 197 197 197 197 197 197 197 ...
    $ mompid   : Factor w/ 21 levels "3587WP.14","3587WP.15",..: 7 7 7 7 7 7 7 7 7 7 ...
    $ dadpid   : Factor w/ 18 levels "3587WP.14","3587WP.7",..: 11 11 11 9 9 9 9 9 9 9 ...
    $ smompop  : Factor w/ 5 levels "hook879WKG","hookWK",..: 1 1 1 1 1 1 1 1 1 1 ...

Data Inspection

Replication

The sample sizes are unbalanced at all levels, including maternal population:

reptab <- with(pol, table(smompop,crosstype))
mosaic(reptab, pop=F)
labeling_cells(text = reptab, margin = 0)(reptab)

Replication is low for some within-population crosses. The replication is even lower for each maternal plant, so we need to be wary of estimates when subsetting at this level:

with(pol, kable(table(mompid,crosstype)))
between within hybrid
3587WP.14 2 0 6
3587WP.15 1 2 1
3587WP.7 3 3 5
3587WP.A 0 2 4
3587WP.C 3 0 0
879WKG.10-1 2 2 12
879WKG.2-2 2 2 17
879WKG.G-2 0 0 6
879WKG.N-5 2 1 0
892WKG.1 5 3 1
892WKG.10 0 0 2
892WKG.3 5 1 3
892WKG.4 0 0 2
892WKG.5 6 3 0
904WPG.2 2 1 10
904WPG.3 3 3 10
904WPG.5 5 2 7
WK.2 10 2 15
WK.2E-1 1 0 7
WK.3 0 1 0
WK.4 5 0 4

Overall data distribution

To identify the best-fitting distribution, we make quantile-quantile plots of the raw data against various distributions. The more points within the confidence interval envelopes, the better the fit. Later, we present quantile-quantile plots of the model residuals to assess model fit.

#QQ plots against various distributions
set.seed(1)
par(mfrow=c(2,4))
normal <- fitdistr(asin(sqrt(pol$vp)), "normal")
qqp(asin(sqrt(pol$vp)), "norm", main="Normal")
lognormal <- fitdistr(asin(sqrt(pol$vp)), "lognormal")
qqp(asin(sqrt(pol$vp)), "lnorm", main="Log Normal")
pois <- fitdistr(asin(sqrt(pol$vp)), "Poisson")
qqp(asin(sqrt(pol$vp)), "pois", pois$estimate, main="Poisson")
gamma <- fitdistr(asin(sqrt(pol$vp)), "gamma")
qqp(asin(sqrt(pol$vp)), "gamma", shape = gamma$estimate[[1]], rate = gamma$estimate[[2]], main="Gamma")

set.seed(1)
par(mfrow=c(2,3))

normal <- fitdistr(log(pol$vp/(1-pol$vp)), "normal")
qqp(log(pol$vp/(1-pol$vp)), "norm", main="Normal")
pois <- fitdistr(log(pol$vp/(1-pol$vp)), "Poisson")
qqp(log(pol$vp/(1-pol$vp)), "pois", pois$estimate, main="Poisson")

distributions by fixed factors

ggplot(pol, aes(x = vp, fill=species)) +
  geom_histogram(data=subset(pol,species == "hook"), aes(y=-..density..),binwidth=0.05)+
  geom_histogram(data=subset(pol,species == "kaal"), aes(y= ..density..),binwidth=0.05)+
  coord_flip() + facet_grid(~crosstype) + labs(y="Histogram", x="Viability")

distributions by random factors

ggplot(aes(y=vp, x=mompid, color=crosstype), data=pol) + geom_count(alpha=0.8) + coord_flip() + labs(x="Maternal plant", y="vp")

Homogeneity of variances across subsets

Our mixed model uses one parameter to capture random effect variance, which is assumed to be homogeneous. Plotting on a log scale should uncouple variances from means to assess this visually. Subsets are species * crosstype * maternal plant.

Subset variances are not homogeneous:

ggplot(aes(y=log(vp+1), x=sxcxm, color=crosstype), data=pol) + geom_boxplot() + coord_flip() + labs(y="ln(Viability + 1)",x="Subsets")

Subset mean-variance relationship

Various distributions make different assumptions about the mean-variance (µ-Var) ratio.

grpVars <- with(pol, tapply(vp, list(sxcxm), var))
grpMeans <- with(pol, tapply(vp, list(sxcxm), mean))
grpCounts <- with(pol, tapply(vp, list(sxcxm), length))
#set weight=grpCounts to weight loess by sample sizes
ggplot(na.omit(data.frame(grpMeans,grpVars,grpCounts)),
       aes(x=grpMeans,y=grpVars, weight=1))+geom_point(aes(size=grpCounts))+
  guides(colour=guide_legend(title="Fit"),size=guide_legend(title="Sample size")) + labs(x="Subset Mean", y="Subset Variance") + labs(subtitle="Subset: species*crosstype*mompid")

Fixed effects

Effects and interactions in these plots are simply given by the mean, which may be unduly influenced by high values.

intplot <- ggplot(pol,aes(x=crosstype,y=vp))+
  geom_count(aes(size = ..prop.., group=sxc),alpha=0.5)+
  stat_summary(aes(x=as.numeric(crosstype)),fun.y=mean,geom="line")+ facet_grid(~species)
intplot + aes(group=species, color=species)

Random effects

Maternal population

intplot + aes(group=mompop, color=mompop)

Maternal plant

intplot + aes(group=mompid, color=mompop)

Paternal population

intplot + aes(group=dadpop, color=dadpop)

Run models on subsets

Run many generalized linear models on subsets of the data defined by crosstype | mompid to see if effects estimates are consistent within maternal plants.

Most maternal plant subsets agree, but some are problematic outliers. These plants can be picked out visually from the random effects interaction plot above, the estimated parameters of each subset model, and the QQ plot of the estimated parameters:

 #had to get rid of species or mompid since mompid is nested inside species. dadpop also works
glm.lis <- lmList(vp~crosstype|mompid,data=pol, family="binomial", weights=pol$tpf)
plot.lmList(glm.lis,scale=list(x=list(relation="free")))
   Loading required package: reshape
   
   Attaching package: 'reshape'
   The following objects are masked from 'package:plyr':
   
       rename, round_any
   The following object is masked from 'package:Matrix':
   
       expand
   Using grp as id variables

qqmath.lmList(glm.lis)#
   Using  as id variables

Models

We constructed the following models with the package glmmADMB. They all have the same fixed effects, species x crosstype, and response variable, vp

  • X = standard GL(M)M
distribution, Random Effects: None Maternal plant Maternal population
normal (norm) X X X 
#Normal (Gaussian) distribution, identity link
#sc.norm          <- glmmadmb(vp~species*crosstype, data=pol, family="gaussian")
#sc.mix.mompid           <- glmmadmb(vp~species*crosstype + (1|mompid), data=pol, family="gaussian")
#sc.mix.mompop    <- glmmadmb(vp~species*crosstype + (1|mompop), data=pol, family="gaussian")

sc.bin          <- glm(vp~species*crosstype, data=pol, family="binomial", weights=pol$tpf)
sc.mix.mompid.bin         <- glmer(vp~species*crosstype + (1|mompid), data=pol, family="binomial", weights=pol$tpf)
sc.mix.mompop.bin    <- glmer(vp~species*crosstype + (1|mompop), data=pol, family="binomial", weights=pol$tpf)

sc.mix.momdadpid.bin         <- glmer(vp~species*crosstype + (1|mompid) + (1|dadpid), data=pol, family="binomial", weights=pol$tpf)

#library(glmmTMB)
#sc.b          <- glmmTMB(vp~species*crosstype, data=pol, family=list(family="beta",link="logit"))
#sc.mix.mompid.b         <- glmmTMB(vp~species*crosstype + (1|mompid), data=pol, family=list(family="beta",link="logit"))
#sc.mix.mompop.b    <- glmmTMB(vp~species*crosstype + (1|mompop), data=pol, family=list(family="beta",link="logit"))

Model comparison

AIC

We will use the Aikake Information Criterion to pick the model the best fits the data, penalized by the number of parameters. Differences of 2 units are significant.

#AICtab(sc.b, sc.mix.mompop.b, sc.mix.mompid.b)
sc.names <- c("sc.bin", "sc.mix.mompid.bin", "sc.mix.mompop.bin","sc.mix.momdadpid.bin")
sc.list <- sapply(sc.names, get, USE.NAMES=T)
sc.AIC <- ICtab(sc.list,mnames=sc.names,type="AIC", base=T, delta=F) # for AICc, nobs=nobs(sc.list[[1]])
class(sc.AIC)<-"data.frame"
all.names <- c(sc.names)
all.list <- sapply(all.names, get, USE.NAMES=T)
all.AIC <- dfun(rbind(sc.AIC))
all.AIC <- all.AIC[order(all.AIC$dAIC),]
kable(all.AIC, format.arg=list(digits=3))
dAIC df
sc.mix.momdadpid.bin 0 8
sc.mix.mompid.bin 66018 7
sc.mix.mompop.bin 108889 7
sc.bin 202551 6

The best-fiting model is a mixed model with the following components:

  • count component (sc.mix.qpoi.tr)
    • response: vp
    • fixed effects: species, crosstype, species x crosstype
    • random effect: mompid

Overdispersion

Looking at the normal, fixed effects model, we see that the residuals are not normal:

shapiro.test(sc.bin$residuals)#raw residuals!
   
    Shapiro-Wilk normality test
   
   data:  sc.bin$residuals
   W = 0.88949, p-value = 7.133e-11

Coefficients

The coefficients estimated for each model agree qualitatively.

sc.log.names <- sc.names
sc.log <- sapply(sc.log.names, get, USE.NAMES=T)

coefplot2(sc.log, legend.x="topright",legend=T,legend.args=list(cex=0.8, xpd=T, inset=c(-0.1,0)), col.pts=sample(gg_color_hue(length(sc.log.names))), spacing=0.05, lwd.2=2, lwd.1=4, intercept=T)

Inference

We chose the model with nearly the best (lowest) AIC, to carry out inference tests and parameter estimation.

Description

mod <- sc.mix.momdadpid.bin
print(mod)
   Generalized linear mixed model fit by maximum likelihood (Laplace
     Approximation) [glmerMod]
    Family: binomial  ( logit )
   Formula: vp ~ species * crosstype + (1 | mompid) + (1 | dadpid)
      Data: pol
   Weights: pol$tpf
         AIC       BIC    logLik  deviance  df.resid 
    305585.6  305611.8 -152784.8  305569.6       189 
   Random effects:
    Groups Name        Std.Dev.
    mompid (Intercept) 0.7222  
    dadpid (Intercept) 0.8569  
   Number of obs: 197, groups:  mompid, 21; dadpid, 18
   Fixed Effects:
                   (Intercept)                  specieskaal  
                        3.2365                      -1.9351  
               crosstypewithin              crosstypehybrid  
                        0.4423                      -3.2530  
   specieskaal:crosstypewithin  specieskaal:crosstypehybrid  
                       -0.2184                       2.7167

Test significance of random effects

Using a likelihood ratio test, with a null hypothesis of zero variance, the random effect (maternal plant) is significant for both model parts:

anova(sc.bin, sc.mix.momdadpid.bin) #double this p-value. or simulate null by permuting data.
   Analysis of Deviance Table
   
   Model: binomial, link: logit
   
   Response: vp
   
   Terms added sequentially (first to last)
   
   
                     Df Deviance Resid. Df Resid. Dev
   NULL                                196     884889
   species            1     9240       195     875649
   crosstype          2   359863       193     515786
   species:crosstype  2     9419       191     506367

Test significance of interaction

By dropping it from the model and performing a likelihood-ratio test, we see that the species x crosstype interaction is not significant.

sxc.chisq <- drop1(mod, test="Chisq") #load from file
dfun(sxc.chisq)
   Single term deletions
   
   Model:
   vp ~ species * crosstype + (1 | mompid) + (1 | dadpid)
                     Df dAIC  LRT   Pr(Chi)    
   <none>                  0                   
   species:crosstype  2 3639 3643 < 2.2e-16 ***
   ---
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model summary

The model estimated the following parameters, with individual parameter significance determined by the Wald z-test, and fixed effect significance determined by analysis of deviance Wald test.

summary(mod)
   Generalized linear mixed model fit by maximum likelihood (Laplace
     Approximation) [glmerMod]
    Family: binomial  ( logit )
   Formula: vp ~ species * crosstype + (1 | mompid) + (1 | dadpid)
      Data: pol
   Weights: pol$tpf
   
         AIC       BIC    logLik  deviance  df.resid 
    305585.6  305611.8 -152784.8  305569.6       189 
   
   Scaled residuals: 
        Min       1Q   Median       3Q      Max 
   -169.337  -16.522    3.018   25.021   93.676 
   
   Random effects:
    Groups Name        Variance Std.Dev.
    mompid (Intercept) 0.5215   0.7222  
    dadpid (Intercept) 0.7343   0.8569  
   Number of obs: 197, groups:  mompid, 21; dadpid, 18
   
   Fixed effects:
                               Estimate Std. Error z value Pr(>|z|)    
   (Intercept)                  3.23651    0.32630    9.92  < 2e-16 ***
   specieskaal                 -1.93509    0.32585   -5.94 2.87e-09 ***
   crosstypewithin              0.44226    0.01653   26.75  < 2e-16 ***
   crosstypehybrid             -3.25305    0.02749 -118.32  < 2e-16 ***
   specieskaal:crosstypewithin -0.21836    0.01839  -11.88  < 2e-16 ***
   specieskaal:crosstypehybrid  2.71672    0.05087   53.40  < 2e-16 ***
   ---
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
   
   Correlation of Fixed Effects:
                   (Intr) spcskl crsstypw crsstyph spcskl:crsstypw
   specieskaal     -0.619                                         
   crsstypwthn     -0.022  0.022                                  
   crsstyphybr     -0.057  0.085  0.210                           
   spcskl:crsstypw  0.017 -0.020 -0.904   -0.155                  
   spcskl:crsstyph  0.054 -0.084 -0.107   -0.973    0.082
Anova(mod, type=3)
   Analysis of Deviance Table (Type III Wald chisquare tests)
   
   Response: vp
                         Chisq Df Pr(>Chisq)    
   (Intercept)          98.383  1  < 2.2e-16 ***
   species              35.267  1  2.874e-09 ***
   crosstype         16781.591  2  < 2.2e-16 ***
   species:crosstype  3117.693  2  < 2.2e-16 ***
   ---
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Predicted random effects

These are box and QQ (to check normality) plots of the estimated random effect of each maternal plant.

predre <- setNames(data.frame(RE=ranef(mod)$mompid,SD=ranef(mod, sd=T)$`1`),c("RE","SD"))
ggplot(predre, aes(x = rownames(predre),y=RE)) +
  geom_point(size = 2) + coord_flip()+
  geom_errorbar(aes(ymin = RE-SD, ymax = RE+SD)) + labs(x="Maternal plants", y="Predicted random effects")

#Count
reStack <- ldply(ranef(mod))
print( qqmath( ~`(Intercept)`|.id, data=reStack, scales=list(relation="free"),
                 prepanel = prepanel.qqmathline,
                 panel = function(x, ...) {
                   panel.qqmathline(x, ...)
                   panel.qqmath(x, ...)
                 },
                 layout=c(1,1)))

Least square means

The least square means procedure can generate predictor estimates of each type, and give their significance groupings with a post-hoc Tukey test. S. hookeri-produced hybrids produce less Viability than either crosses between or within S. hookeri populations. The other differences are not significant, but remember that the fixed effect of hybrid (vs. between) was significant (model summary).

#Count
rg <- ref.grid(mod)
#summary(rg)
sxc.lsm <- lsmeans(rg, ~ crosstype*species)
plot(sxc.lsm)

cld.mod <- cld(sxc.lsm, Letters=letters) #tukey letterings
library(boot)
   
   Attaching package: 'boot'
   The following object is masked from 'package:lattice':
   
       melanoma
   The following object is masked from 'package:car':
   
       logit
cld.mod$response <- inv.logit(cld.mod$lsmean)
options(digits=4)
cld.mod[rev(order(cld.mod$species, cld.mod$crosstype)),]
    crosstype species   lsmean     SE df asymp.LCL asymp.UCL .group response
    hybrid    kaal     0.76509 0.2848 NA    0.2068    1.3234  a       0.6825
    within    kaal     1.52532 0.2847 NA    0.9673    2.0833    c     0.8213
    between   kaal     1.30142 0.2846 NA    0.7435    1.8593   b      0.7861
    hybrid    hook    -0.01654 0.3259 NA   -0.6553    0.6222  a       0.4959
    within    hook     3.67877 0.3264 NA    3.0391    4.3184      e   0.9754
    between   hook     3.23651 0.3263 NA    2.5970    3.8760     d    0.9622
   
   Results are given on the logit (not the response) scale. 
   Confidence level used: 0.95 
   Results are given on the log odds ratio (not the response) scale. 
   P value adjustment: tukey method for comparing a family of 6 estimates 
   significance level used: alpha = 0.05