Descriptive Statistics
IQR
I implemented function IQrange before the standard function IQR gained the type argument. Since 2010 (r53643, r53644) the function is identical to function IQR.
## [1] 1.1782
## [1] 1.1782
It is also possible to compute a standardized version of the IQR leading to a normal-consistent estimate of the standard deviation.
## [1] 0.873401
## [1] 0.9414619
Mean Absolute Deviation
The mean absolute deviation under the assumption of symmetry is a robust alternative to the sample standard deviation.
## [1] 0.9111063
Five Number Summary
There is a function that computes a so-called five number summary which in contrast to function fivenum uses the first and third quartile instead of the lower and upper hinge.
## Minimum 25% Median 75% Maximum
## -3.05836109 -0.66319183 -0.04731851 0.51500822 2.19052837
Coefficient of Variation (CV)
There are functions to compute the (classical) coefficient of variation as well as two robust variants. In case of the robust variants, the mean is replaced by the median and the SD is replaced by the (standardized) MAD and the (standardized) IQR, respectively.
## [1] 7
## [1] 0.409788
## [1] 0.243109
## [1] 0.2525377
Signal to Noise Ratio (SNR)
There are functions to compute the (classical) signal to noise ratio as well as two robust variants. In case of the robust variants, the mean is replaced by the median and the SD is replaced by the (standardized) MAD and the (standardized) IQR, respectively.
## [1] 2.440286
## [1] 4.113381
## [1] 3.959805
Box- and Whisker-Plot
In contrast to the standard function boxplot which uses the lower and upper hinge for defining the box and the whiskers, the function qboxplot uses the first and third quartile.
The difference between the two versions often is hardly visible.
OR, RR and Other Risk Measures
Given the incidence of the outcome of interest in the nonexposed (p0) and exposed (p1) group, several risk measures can be computed.
## p0 p1 RR OR RRR ARR NNT
## 0.4000000 0.1000000 0.2500000 0.1666667 0.7500000 0.3000000 3.3333333
## p0 p1 RR OR RRI ARI NNH
## 0.40 0.50 1.25 1.50 0.25 0.10 10.00
Given p0 or p1 and OR, we can compute the respective RR.
## [1] 1.25
## [1] 0.25
Generalized Logarithm
The generalized logarithm may be useful as a variance stabilizing transformation when also negative values are present.
## Warning in log(x): NaNs wurden erzeugt
As in case of function log there is also glog10 and glog2.
## Warning in eval(expr, envir = ll, enclos = parent.frame()): NaNs wurden erzeugt
There are also functions that compute the inverse of the generalized logarithm.
## [1] 10
## [1] 10
## [1] 10
## [1] 10
Plot TSH, fT3 and fT4 Values
The thyroid function is usually investigated by determining the values of TSH, fT3 and fT4. The function thyroid can be used to visualize the measured values as relative values with respect to the provided reference ranges.
Change Data from Wide to Long
Often it’s better to have the data in a long format than in a wide format; e.g., when plotting with package ggplot2. The necessary transformation can be done with function melt.long.
## value variable
## x1 1.11524309 x
## x2 0.94089105 x
## x3 -2.13757591 x
## x4 -0.67961279 x
## x5 1.30650488 x
## x6 1.33565454 x
## x7 0.48937445 x
## x8 0.05582566 x
## x9 0.22164920 x
## x10 -1.28249794 x
## y1 1.14605275 y
## y2 -1.23622737 y
## y3 -0.79443916 y
## y4 -0.66390875 y
## y5 1.66328198 y
## y6 -0.26933457 y
## y7 -0.70839818 y
## y8 -1.25013804 y
## y9 0.18558331 y
## y10 -0.54337164 y
## z1 -0.73433758 z
## z2 0.19908900 z
## z3 -2.12946669 z
## z4 2.33596080 z
## z5 -0.92474633 z
## z6 1.70552543 z
## z7 -0.10159526 z
## z8 -0.38834693 z
## z9 -1.33659791 z
## z10 0.54298135 z
## group value variable
## x1 a 1.11524309 x
## x2 a 0.94089105 x
## x3 a -2.13757591 x
## x4 a -0.67961279 x
## x5 a 1.30650488 x
## x6 b 1.33565454 x
## x7 b 0.48937445 x
## x8 b 0.05582566 x
## x9 b 0.22164920 x
## x10 b -1.28249794 x
## y1 a 1.14605275 y
## y2 a -1.23622737 y
## y3 a -0.79443916 y
## y4 a -0.66390875 y
## y5 a 1.66328198 y
## y6 b -0.26933457 y
## y7 b -0.70839818 y
## y8 b -1.25013804 y
## y9 b 0.18558331 y
## y10 b -0.54337164 y
Confidence Intervals
Binomial Proportion
There are several functions for computing confidence intervals. We can compute 10 different confidence intervals for binomial proportions; e.g.
##
## wilson confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## prob 0.1429739 0.3741268
##
## sample estimate:
## prob
## 0.24
##
## additional information:
## standard error of prob
## 0.05896867
##
## clopper-pearson confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## prob 0.1306099 0.3816907
##
## sample estimate:
## prob
## 0.24
## [1] 0.1306099 0.3816907
## attr(,"conf.level")
## [1] 0.95
For all intervals implemented see the help page of function binomCI.
Mean and SD
We can compute confidence intervals for mean and SD of a normal distribution.
##
## Exact confidence interval(s)
##
## 95 percent confidence intervals:
## 2.5 % 97.5 %
## mean 1.469105 3.203649
## sd 2.549156 3.802776
##
## sample estimates:
## mean sd
## 2.336377 3.051661
##
## additional information:
## SE of mean
## 0.43157
##
## Exact confidence interval(s)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## mean 1.504835 3.167919
##
## sample estimate:
## mean
## 2.336377
##
## additional information:
## SE of mean
## 0.4242641
##
## Exact confidence interval(s)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## sd 2.549156 3.802776
##
## sample estimate:
## sd
## 3.051661
Difference in Means
We can compute confidence interval for the difference of means assuming normal distributions.
##
## Confidence interval (paired)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## mean of differences -0.581584 1.095012
##
## sample estimate:
## mean of differences
## 0.2567139
##
## additional information:
## SE of mean of differences
## 0.40052
##
## Exact confidence interval(s)
##
## 95 percent confidence intervals:
## 2.5 % 97.5 %
## mean -0.581584 1.095012
## sd 1.362175 2.616146
##
## sample estimates:
## mean sd
## 0.2567139 1.7911797
##
## additional information:
## SE of mean
## 0.40052
##
## Classical confidence interval (unpaired)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## difference in means -3.193029 -0.8745933
##
## sample estimate:
## difference in means
## -2.033811
##
## additional information:
## SE of difference in means Cohen's d (SMD)
## 0.5659119 -1.3918981
##
## mean of x SD of x mean of y SD of y
## 0.1266118 1.0522706 2.1604231 2.0747952
##
## Welch confidence interval (unpaired)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## difference in means -3.561805 -0.5058175
##
## sample estimate:
## difference in means
## -2.033811
##
## additional information:
## SE of difference in means Cohen's d (SMD)
## 0.6970231 -0.8742382
##
## mean of x SD of x mean of y SD of y
## 0.1266118 1.0522706 2.1604231 2.0747952
##
## Hsu confidence interval (unpaired)
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## difference in means -3.610587 -0.4570355
##
## sample estimate:
## difference in means
## -2.033811
##
## additional information:
## SE of difference in means Cohen's d (SMD)
## 0.6970231 -0.8742382
##
## mean of x SD of x mean of y SD of y
## 0.1266118 1.0522706 2.1604231 2.0747952
In case of unequal variances and unequal sample sizes per group the classical confidence interval may have a bad coverage (too long or too short), as is indicated by the small Monte-Carlo simulation study below.
M <- 100
CIhsu <- CIwelch <- CIclass <- matrix(NA, nrow = M, ncol = 2)
for(i in 1:M){
x <- rnorm(10)
y <- rnorm(30, sd = 0.1)
CIclass[i,] <- normDiffCI(x, y, method = "classical")$conf.int
CIwelch[i,] <- normDiffCI(x, y, method = "welch")$conf.int
CIhsu[i,] <- normDiffCI(x, y, method = "hsu")$conf.int
}
## coverage probabilies
## classical
sum(CIclass[,1] < 0 & 0 < CIclass[,2])/M
## [1] 0.74
## [1] 0.95
## [1] 0.95
Coefficient of Variation
We provide 11 different confidence intervals for the (classical) coefficient of variation; e.g.
##
## Miller (1991) confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## CV 0.1682293 0.2250942
##
## sample estimate:
## CV
## 0.1966618
##
## additional information:
## standard error of CV
## 0.01450661
##
## Gulhar et al (2012) confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## CV 0.1726704 0.2284571
##
## sample estimate:
## CV
## 0.1966618
For all intervals implemented see the help page of function cvCI.
Quantiles, Median and MAD
We start with the computation of confidence intervals for quantiles.
##
## exact confidence interval
##
## 95.14464 percent confidence interval:
## lower upper
## 95 % quantile 4.359203 6.336234
##
## sample estimate:
## 95 % quantile
## 5.155109
##
## asymptotic confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## 95 % quantile 4.359203 6.649789
##
## sample estimate:
## 95 % quantile
## 5.155109
Next, we consider the median.
##
## exact confidence interval
##
## 95.23684 percent confidence intervals:
## lower upper
## median 0.703545 1.646463
## median 1.019803 1.969207
##
## sample estimate:
## median
## 1.303773
##
## asymptotic confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## median 0.86415 1.749188
##
## sample estimate:
## median
## 1.303773
It often happens that quantile confidence intervals are not unique. Here the minimum length interval might be of interest.
##
## minimum length exact confidence interval
##
## 95.23684 percent confidence interval:
## lower upper
## median 0.703545 1.646463
##
## sample estimate:
## median
## 1.303773
Finally, we take a look at MAD (median absolute deviation) where by default the standardized MAD is used (see function mad).
##
## exact confidence interval
##
## 95.23684 percent confidence intervals:
## lower upper
## MAD 1.180799 1.631619
## MAD 1.282536 1.694431
##
## sample estimate:
## MAD
## 1.503738
##
## asymptotic confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## MAD 1.216314 1.638922
##
## sample estimate:
## MAD
## 1.503738
##
## exact confidence interval
##
## 95.23684 percent confidence intervals:
## lower upper
## MAD 0.7964378 1.100512
## MAD 0.8650585 1.142878
##
## sample estimate:
## MAD
## 1.014257
Relative Risk
There is also a function for computing an approximate confidence interval for the relative risk (RR).
##
## asymptotic confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## relative risk 0.1510993 0.4136353
##
## sample estimate:
## relative risk
## 0.25
##
## asymptotic confidence interval
##
## 95 percent confidence interval:
## 2.5 % 97.5 %
## relative risk 1.00256 1.55851
##
## sample estimate:
## relative risk
## 1.25
Sample Size
Welch Two-Sample t-Test
For computing the sample size of the Welch t-test, we only consider the situation of equal group size (balanced design).
##
## Two-sample Welch t test power calculation
##
## n = 20
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.8689528
## alternative = two.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Welch t test power calculation
##
## n = 22.0211
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Welch t test power calculation
##
## n = 17.84713
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = one.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Welch t test power calculation
##
## n = 14.53583
## delta = 1
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
Hsu Two-Sample t-Test
For computing the sample size of the Hsu t-test, we only consider the situation of equal group size (balanced design).
##
## Two-sample Hsu t test power calculation
##
## n = 20
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.8506046
## alternative = two.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Hsu t test power calculation
##
## n = 23.02186
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Hsu t test power calculation
##
## n = 18.5674
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = one.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Welch t test power calculation
##
## n = 53.86822
## delta = 0.5
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
##
## Two-sample Hsu t test power calculation
##
## n = 54.49421
## delta = 0.5
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
Two Negative Binomial Rates
When we consider two negative binomial rates, we can compute sample size or power applying function power.nb.test.
##
## Power calculation for comparing two negative binomial rates
##
## n = 22.37386
## n1 = 22.37386
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
##
## Power calculation for comparing two negative binomial rates
##
## n = 21.23734
## n1 = 21.23734
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
##
## Power calculation for comparing two negative binomial rates
##
## n = 20.85564
## n1 = 20.85564
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
Moderated Tests Based on Package limma
Moderated t-Test
The function to compute the moderated t-test was motivated by the fact that my students have problems to understand and correctly adapt the code of the limma package.
## mean 2.5% 97.5% t p.value adj.p.value B
## 1 1.1459007 0.7563349 1.535467 5.865241 1.306474e-07 4.354915e-07 7.2129144
## 2 1.2450439 0.8365495 1.653538 6.077407 5.501014e-08 2.750507e-07 8.0521639
## 3 0.8207214 0.4347155 1.206727 4.239567 6.642097e-05 8.302621e-05 1.2167884
## 4 0.7056067 0.2791414 1.132072 3.299125 1.518876e-03 1.518876e-03 -1.7280496
## 5 1.1031323 0.6723651 1.533900 5.106281 2.651347e-06 5.314983e-06 4.3025833
## 6 1.1485864 0.7740776 1.523095 6.115351 4.708411e-08 2.750507e-07 8.2032276
## 7 0.9512675 0.5797591 1.322776 5.105682 2.657492e-06 5.314983e-06 4.3003533
## 8 0.8963567 0.4879264 1.304787 4.376054 4.075468e-05 5.822097e-05 1.6820734
## 9 0.8026077 0.3983796 1.206836 3.959101 1.766913e-04 1.963237e-04 0.2886083
## 10 0.8850387 0.4918235 1.278254 4.487988 2.714895e-05 4.524824e-05 2.0698891
## Two-sample test
set.seed(123)
X <- rbind(matrix(rnorm(5*20), nrow = 5, ncol = 20),
matrix(rnorm(5*20, mean = 1), nrow = 5, ncol = 20))
g2 <- factor(c(rep("group 1", 10), rep("group 2", 10)))
mod.t.test(X, group = g2)
## difference in means 2.5% 97.5% t p.value adj.p.value
## 1 -0.30099317 -1.1337943 0.5318079 -0.71317027 0.4766640 0.9665915
## 2 0.13637650 -0.6964246 0.9691776 0.32312914 0.7469725 0.9665915
## 3 -0.49260925 -1.3254104 0.3401919 -1.16718354 0.2446801 0.8156005
## 4 0.15266680 -0.6801343 0.9854679 0.36172723 0.7179801 0.9665915
## 5 -0.05546448 -0.8882656 0.7773366 -0.13141699 0.8955922 0.9665915
## 6 -0.03152278 -0.8643239 0.8012783 -0.07468976 0.9405445 0.9665915
## 7 -0.69311916 -1.5259203 0.1396820 -1.64226977 0.1022800 0.8156005
## 8 -0.17595066 -1.0087518 0.6568505 -0.41689577 0.6772514 0.9665915
## 9 -0.54524220 -1.3780433 0.2875589 -1.29189154 0.1980510 0.8156005
## 10 -0.01770155 -0.8505027 0.8150996 -0.04194188 0.9665915 0.9665915
## B mean of group 1 mean of group 2
## 1 -4.608912 0.07083562 0.371828786
## 2 -4.619655 0.13562592 -0.000750575
## 3 -4.586223 -0.30436677 0.188242477
## 4 -4.618953 0.18497579 0.032308986
## 5 -4.621971 0.08494719 0.140411667
## 6 -4.622282 1.11828347 1.149806246
## 7 -4.550748 0.67404426 1.367163425
## 8 -4.617811 0.53111339 0.707064052
## 9 -4.578072 0.92210638 1.467348578
## 10 -4.622384 0.48495033 0.502651884
## mean of differences 2.5% 97.5% t p.value
## 1 -0.30099317 -0.7260027 0.12401634 -1.39744874 0.163998732
## 2 0.13637650 -0.2886330 0.56138600 0.63316773 0.527427738
## 3 -0.49260925 -0.9176188 -0.06759974 -2.28708237 0.023354856
## 4 0.15266680 -0.2723427 0.57767631 0.70880025 0.479365424
## 5 -0.05546448 -0.4804740 0.36954503 -0.25751005 0.797079211
## 6 -0.03152278 -0.4565323 0.39348673 -0.14635370 0.883805952
## 7 -0.69311916 -1.1181287 -0.26810966 -3.21800824 0.001531067
## 8 -0.17595066 -0.6009602 0.24905885 -0.81690235 0.415064109
## 9 -0.54524220 -0.9702517 -0.12023269 -2.53144623 0.012215144
## 10 -0.01770155 -0.4427111 0.40730796 -0.08218462 0.934591222
## adj.p.value B
## 1 0.40999683 -5.193223
## 2 0.75346820 -5.932720
## 3 0.07784952 -3.631339
## 4 0.75346820 -5.884358
## 5 0.93459122 -6.092149
## 6 0.93459122 -6.113540
## 7 0.01531067 -1.189400
## 8 0.75346820 -5.805769
## 9 0.06107572 -3.070283
## 10 0.93459122 -6.120528
Moderated 1-Way ANOVA
The function to compute a moderated 1-way ANOVA was motivated by the fact that my students have problems to understand and correctly adapt the code of the limma package.
set.seed(123)
X <- rbind(matrix(rnorm(5*20), nrow = 5, ncol = 20),
matrix(rnorm(5*20, mean = 1), nrow = 5, ncol = 20))
gr <- factor(c(rep("A1", 5), rep("B2", 5), rep("C3", 5), rep("D4", 5)))
mod.oneway.test(X, gr)
## topTableF is obsolete and will be removed in a future version of limma. Please considering using topTable instead.
## A-B A-C A-D B-C B-D C-D
## 1 1.0974330 0.2271669 0.26827976 -0.87026609 -0.8291532 0.04111286
## 2 0.0769194 0.9622020 -0.61252965 0.88528264 -0.6894491 -1.57473169
## 3 -0.3105477 -0.7130963 -0.58266984 -0.40254866 -0.2721221 0.13042651
## 4 -0.5832304 -0.2440290 -0.03386782 0.33920142 0.5493626 0.21016117
## 5 -0.9579128 -0.4501003 -0.61874146 0.50781250 0.3391713 -0.16864116
## 6 -0.6013595 -0.5109767 -0.15342834 0.09038278 0.4479311 0.35754833
## 7 0.3471695 -0.3493145 -0.68975433 -0.69648400 -1.0369239 -0.34043986
## 8 -0.9277388 -0.5691081 -0.71053196 0.35863064 0.2172068 -0.14142382
## 9 -0.5998638 -1.5488157 -0.14153255 -0.94895185 0.4583313 1.40728311
## 10 1.5527067 0.5179562 0.99934740 -1.03475050 -0.5533593 0.48139118
## grand mean F p.value adj.p.value
## 1 0.22133220 1.4119467 0.23709800 0.5868380
## 2 0.06743767 2.5678204 0.05255479 0.1751826
## 3 -0.05806215 0.6094415 0.60879068 0.7253749
## 4 0.10864239 0.4386138 0.72537489 0.7253749
## 5 0.11267943 0.9691456 0.40611786 0.5868380
## 6 1.13404486 0.5002131 0.68212276 0.7253749
## 7 1.02060384 1.2185746 0.30111975 0.5868380
## 8 0.61908872 0.9594030 0.41078662 0.5868380
## 9 1.19472748 2.9871234 0.02980893 0.1751826
## 10 0.49380111 2.6903326 0.04456696 0.1751826
Pairwise moderated t-tests
As a optional post-hoc analysis after mod.oneway.test one can use pairwise moderated t-tests. One should carefully think about the adjustment of p values in this context.
## A1-B2 A1-B2: t A1-B2: p A1-B2: adj.p A1-B2: B A1-C3
## 1 1.0974330 1.9176595 0.056935659 0.26739434 -4.150296 0.2271669
## 2 0.0769194 0.1344093 0.893247840 0.89324784 -5.826707 0.9622020
## 3 -0.3105477 -0.5426525 0.588124185 0.65347132 -5.700081 -0.7130963
## 4 -0.5832304 -1.0191395 0.309675276 0.44239325 -5.359162 -0.2440290
## 5 -0.9579128 -1.6738612 0.096111665 0.26739434 -4.551426 -0.4501003
## 6 -0.6013595 -1.0508183 0.294926580 0.44239325 -5.329122 -0.5109767
## 7 0.3471695 0.6066456 0.544945915 0.65347132 -5.666387 -0.3493145
## 8 -0.9277388 -1.6211350 0.106957735 0.26739434 -4.631016 -0.5691081
## 9 -0.5998638 -1.0482048 0.296125067 0.44239325 -5.331635 -1.5488157
## 10 1.5527067 2.7132069 0.007393498 0.07393498 -2.462559 0.5179562
## A1-C3: t A1-C3: p A1-C3: adj.p A1-C3: B A1-D4 A1-D4: t
## 1 0.3969525 0.691931480 0.69193148 -5.757494 0.26827976 0.46879329
## 2 1.6813563 0.094644661 0.47322331 -4.535810 -0.61252965 -1.07033714
## 3 -1.2460679 0.214560903 0.61819096 -5.119000 -0.58266984 -1.01815997
## 4 -0.4264174 0.670376894 0.69193148 -5.746391 -0.03386782 -0.05918078
## 5 -0.7865073 0.432733671 0.61819096 -5.546500 -0.61874146 -1.08119168
## 6 -0.8928830 0.373260796 0.61819096 -5.464740 -0.15342834 -0.26810139
## 7 -0.6103937 0.542467061 0.67808383 -5.659092 -0.68975433 -1.20527990
## 8 -0.9944622 0.321499481 0.61819096 -5.376999 -0.71053196 -1.24158683
## 9 -2.7064076 0.007539705 0.07539705 -2.477381 -0.14153255 -0.24731463
## 10 0.9050791 0.366784330 0.61819096 -5.454705 0.99934740 1.74626427
## A1-D4: p A1-D4: adj.p A1-D4: B B2-C3 B2-C3: t B2-C3: p
## 1 0.63985549 0.8944242 -4.606903 -0.87026609 -1.5207070 0.13030713
## 2 0.28607955 0.5168984 -4.593186 0.88528264 1.5469470 0.12385169
## 3 0.31013903 0.5168984 -4.594801 -0.40254866 -0.7034154 0.48281925
## 4 0.95288197 0.9528820 -4.610107 0.33920142 0.5927221 0.55420392
## 5 0.28123878 0.5168984 -4.592840 0.50781250 0.8873539 0.37622030
## 6 0.78896661 0.8944242 -4.609094 0.09038278 0.1579353 0.87470693
## 7 0.22987482 0.5168984 -4.588637 -0.69648400 -1.2170394 0.22538132
## 8 0.21620608 0.5168984 -4.587321 0.35863064 0.6266728 0.53176683
## 9 0.80498180 0.8944242 -4.609252 -0.94895185 -1.6582029 0.09923575
## 10 0.08268426 0.5168984 -4.564982 -1.03475050 -1.8081278 0.07246459
## B2-C3: adj.p B2-C3: B B2-D4 B2-D4: t B2-D4: p B2-D4: adj.p
## 1 0.3257678 -4.575899 -0.8291532 -1.4488662 0.14933196 0.6213576
## 2 0.3257678 -4.574706 -0.6894491 -1.2047465 0.23008017 0.6213576
## 3 0.6157821 -4.602828 -0.2721221 -0.4755075 0.63507382 0.7047841
## 4 0.6157821 -4.604954 0.5493626 0.9599587 0.33852439 0.6213576
## 5 0.6157821 -4.598493 0.3391713 0.5926696 0.55423902 0.6927988
## 6 0.8747069 -4.609789 0.4479311 0.7827169 0.43495030 0.6213576
## 7 0.4507626 -4.588215 -1.0369239 -1.8119255 0.07187282 0.6213576
## 8 0.6157821 -4.604340 0.2172068 0.3795482 0.70478410 0.7047841
## 9 0.3257678 -4.569423 0.4583313 0.8008902 0.42438292 0.6213576
## 10 0.3257678 -4.561724 -0.5533593 -0.9669426 0.33503202 0.6213576
## B2-D4: B C3-D4 C3-D4: t C3-D4: p C3-D4: adj.p C3-D4: B
## 1 -4.579059 0.04111286 0.0718408 0.942818328 0.94281833 -5.862037
## 2 -4.588656 -1.57473169 -2.7516934 0.006613244 0.06613244 -2.377508
## 3 -4.606809 0.13042651 0.2279079 0.820008840 0.91112093 -5.840494
## 4 -4.596506 0.21016117 0.3672366 0.713927813 0.91112093 -5.802308
## 5 -4.604955 -0.16864116 -0.2946843 0.768617018 0.91112093 -5.824423
## 6 -4.601082 0.35754833 0.6247816 0.533004365 0.91112093 -5.684652
## 7 -4.561521 -0.34043986 -0.5948862 0.552759917 0.91112093 -5.701444
## 8 -4.608024 -0.14142382 -0.2471246 0.805128573 0.91112093 -5.836290
## 9 -4.600656 1.40728311 2.4590930 0.014993757 0.07496878 -3.079639
## 10 -4.596307 0.48139118 0.8411852 0.401499618 0.91112093 -5.538559
Hsu Two-Sample t-Test
The Hsu two-sample t-test is an alternative to the Welch two-sample t-test using a different formula for computing the degrees of freedom of the respective t-distribution. The following code is taken and adapted from the help page of the t.test function.
##
## Welch Two Sample t-test
##
## data: 1:10 and c(7:20)
## t = -5.4349, df = 21.982, p-value = 1.855e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.052802 -4.947198
## sample estimates:
## mean of x mean of y
## 5.5 13.5
##
## Welch Two Sample t-test
##
## data: 1:10 and c(7:20, 200)
## t = -1.6329, df = 14.165, p-value = 0.1245
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -47.242900 6.376233
## sample estimates:
## mean of x mean of y
## 5.50000 25.93333
##
## Hsu Two Sample t-test
##
## data: 1:10 and c(7:20)
## t = -5.4349, df = 9, p-value = 0.0004137
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.329805 -4.670195
## sample estimates:
## mean of x mean of y SD of x SD of y
## 5.50000 13.50000 3.02765 4.18330
##
## Hsu Two Sample t-test
##
## data: 1:10 and c(7:20, 200)
## t = -1.6329, df = 9, p-value = 0.1369
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -48.740846 7.874179
## sample estimates:
## mean of x mean of y SD of x SD of y
## 5.50000 25.93333 3.02765 48.32253
##
## Welch Two Sample t-test
##
## data: extra[group == 1] and extra[group == 2]
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -3.3654832 0.2054832
## sample estimates:
## mean of x mean of y
## 0.75 2.33
##
## Hsu Two Sample t-test
##
## data: extra[group == 1] and extra[group == 2]
## t = -1.8608, df = 9, p-value = 0.09569
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -3.5007773 0.3407773
## sample estimates:
## mean of x mean of y SD of x SD of y
## 0.750000 2.330000 1.789010 2.002249
##
## Welch Two Sample t-test
##
## data: extra by group
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means between group 1 and group 2 is not equal to 0
## 95 percent confidence interval:
## -3.3654832 0.2054832
## sample estimates:
## mean in group 1 mean in group 2
## 0.75 2.33
##
## Hsu Two Sample t-test
##
## data: extra by group
## t = -1.8608, df = 9, p-value = 0.09569
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -3.5007773 0.3407773
## sample estimates:
## mean of x mean of y SD of x SD of y
## 0.750000 2.330000 1.789010 2.002249
Multiple Imputation t-Test
Function mi.t.test can be used to compute a multiple imputation t-test by applying the approch of Rubin (1987) in combination with the adjustment of Barnard and Rubin (1999).
## Generate some data
set.seed(123)
x <- rnorm(25, mean = 1)
x[sample(1:25, 5)] <- NA
y <- rnorm(20, mean = -1)
y[sample(1:20, 4)] <- NA
pair <- c(rnorm(25, mean = 1), rnorm(20, mean = -1))
g <- factor(c(rep("yes", 25), rep("no", 20)))
D <- data.frame(ID = 1:45, variable = c(x, y), pair = pair, group = g)
## Use Amelia to impute missing values
library(Amelia)
## Lade nötiges Paket: Rcpp
## ##
## ## Amelia II: Multiple Imputation
## ## (Version 1.8.0, built: 2021-05-26)
## ## Copyright (C) 2005-2022 James Honaker, Gary King and Matthew Blackwell
## ## Refer to http://gking.harvard.edu/amelia/ for more information
## ##
##
## Welch Two Sample t-test
##
## data: variable by group
## t = -6.9214, df = 33.69, p-value = 5.903e-08
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
## -2.628749 -1.435125
## sample estimates:
## mean in group no mean in group yes
## -1.0862469 0.9456901
##
## Multiple Imputation Welch Two Sample t-test
##
## data: Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 5.63e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.668011 -1.405862
## sample estimates:
## mean (no) SD (no) mean (yes) SD (yes)
## -1.0809584 0.9379070 0.9559786 1.0203219
##
## Two Sample t-test
##
## data: variable by group
## t = -6.8168, df = 34, p-value = 7.643e-08
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
## -2.637703 -1.426171
## sample estimates:
## mean in group no mean in group yes
## -1.0862469 0.9456901
##
## Multiple Imputation Two Sample t-test
##
## data: Variable variable: group no vs group yes
## t = -6.5736, df = 26.042, p-value = 5.656e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.673828 -1.400046
## sample estimates:
## mean (no) SD (no) mean (yes) SD (yes)
## -1.0809584 0.9379070 0.9559786 1.0203219
##
## Multiple Imputation Welch Two Sample t-test
##
## data: Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 2.815e-07
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
## -Inf -1.5135
## sample estimates:
## mean (no) SD (no) mean (yes) SD (yes)
## -1.0809584 0.9379070 0.9559786 1.0203219
##
## Multiple Imputation Welch Two Sample t-test
##
## data: Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 1
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## -2.560374 Inf
## sample estimates:
## mean (no) SD (no) mean (yes) SD (yes)
## -1.0809584 0.9379070 0.9559786 1.0203219
##
## One Sample t-test
##
## data: D$variable[D$group == "yes"]
## t = 4.5054, df = 19, p-value = 0.0002422
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 0.5063618 1.3850184
## sample estimates:
## mean of x
## 0.9456901
##
## Multiple Imputation One Sample t-test
##
## data: Variable variable
## t = 4.6847, df = 18.494, p-value = 0.0001725
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 0.5280752 1.3838820
## sample estimates:
## mean SD
## 0.9559786 1.0203219
##
## Multiple Imputation One Sample t-test
##
## data: Variable variable
## t = 9.5851, df = 18.494, p-value = 1
## alternative hypothesis: true mean is less than -1
## 95 percent confidence interval:
## -Inf 1.309328
## sample estimates:
## mean SD
## 0.9559786 1.0203219
##
## Multiple Imputation One Sample t-test
##
## data: Variable variable
## t = 9.5851, df = 18.494, p-value = 6.655e-09
## alternative hypothesis: true mean is greater than -1
## 95 percent confidence interval:
## 0.6026297 Inf
## sample estimates:
## mean SD
## 0.9559786 1.0203219
##
## Paired t-test
##
## data: D$variable and D$pair
## t = -1.3532, df = 35, p-value = 0.1847
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
## -0.6976921 0.1395993
## sample estimates:
## mean difference
## -0.2790464
##
## Multiple Imputation Paired t-test
##
## data: Variables variable and pair
## t = -1.0455, df = 39.974, p-value = 0.3021
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.5680741 0.1807237
## sample estimates:
## mean of difference SD of difference
## -0.1936752 1.2426515
Imputation of Standard Deviations for Changes from Baseline
The function imputeSD can be used to impute standard deviations for changes from baseline adopting the approach of Section 16.1.3.2 of the Cochrane handbook (2011).
SD1 <- c(0.149, 0.022, 0.036, 0.085, 0.125, NA, 0.139, 0.124, 0.038)
SD2 <- c(NA, 0.039, 0.038, 0.087, 0.125, NA, 0.135, 0.126, 0.038)
SDchange <- c(NA, NA, NA, 0.026, 0.058, NA, NA, NA, NA)
imputeSD(SD1, SD2, SDchange)
## SD1 SD2 SDchange Cor SDchange.min SDchange.mean SDchange.max
## 1 0.149 0.149 NA NA 0.04491608 0.05829769 0.06913600
## 2 0.022 0.039 NA NA 0.01915642 0.02050235 0.02176520
## 3 0.036 0.038 NA NA 0.01132754 0.01460887 0.01727787
## 4 0.085 0.087 0.026 0.9545639 0.02600000 0.02600000 0.02600000
## 5 0.125 0.125 0.058 0.8923520 0.05800000 0.05800000 0.05800000
## 6 NA NA NA NA NA NA NA
## 7 0.139 0.135 NA NA 0.04148755 0.05374591 0.06368696
## 8 0.124 0.126 NA NA 0.03773311 0.04894677 0.05803262
## 9 0.038 0.038 NA NA 0.01145511 0.01486787 0.01763200
AUC
Estimation
There are two functions that can be used to calculate and test AUC values. First function AUC, which computes the area under the receiver operating characteristic curve (AUC under ROC curve) using the connection of AUC to the Wilcoxon rank sum test. We use some random data and groups to demonstrate the use of this function.
## [1] 0.9212
Sometimes the labels of the group should be switched to avoid an AUC smaller than 0.5, which represents a result worse than a pure random choice.
## Warning in AUC(x, group = g): The computed AUC value 0.0788 will be replaced by
## 0.9212 which can be achieved be interchanging the sample labels!
## [1] 0.9212
## [1] 0.0788
Testing
We can also perform statistical tests for AUC. First, the one-sample test which corresponds to the Wilcoxon signed rank test.
## $Variable1
## AUC SE low CI up CI
## 0.9212000 0.0285512 0.8652407 0.9771593
##
## $Test
##
## Wilcoxon rank sum test with continuity correction
##
## data: 0 and 1
## W = 197, p-value = 3.995e-13
## alternative hypothesis: true AUC is not equal to 0.5
We can also compare two AUC using the test of Hanley and McNeil (1982).
## $Variable1
## AUC SE low CI up CI
## 0.9212000 0.0285512 0.8652407 0.9771593
##
## $Variable2
## AUC SE low CI up CI
## 0.78560000 0.04585469 0.69572646 0.87547354
##
## $Test
##
## Hanley and McNeil test for two AUCs
##
## data: x and x2
## z = 2.5103, p-value = 0.01206
## alternative hypothesis: true difference in AUC is not equal to 0
## 95 percent confidence interval:
## 0.02972886 0.24147114
## sample estimates:
## Difference in AUC
## 0.1356
Pairwise
There is also a function for pairwise comparison if there are more than two groups.
## Warning in AUC(xi, xj): The computed AUC value 0.0788 will be replaced by 0.9212
## which can be achieved be interchanging the sample labels!
## Warning in AUC(xi, xj): The computed AUC value 0.4076 will be replaced by 0.5924
## which can be achieved be interchanging the sample labels!
## Warning in AUC(xi, xj): The computed AUC value 0.1588 will be replaced by 0.8412
## which can be achieved be interchanging the sample labels!
## Warning in AUC(xi, xj): The computed AUC value 0.2144 will be replaced by 0.7856
## which can be achieved be interchanging the sample labels!
## 0 vs 1 0 vs 2 0 vs 3 1 vs 2 1 vs 3 2 vs 3
## 0.9212 0.5924 0.8412 0.8992 0.6568 0.7856
In addition to the pairwise.auc there are further functions for pairwise comparisons.
Pairwise Comparisons
Often we are in a situation that we want to compare more than two groups pairwise.
FC and logFC
In the analysis of omics data, the FC or logFC are important measures and are often used in combination with (adjusted) p values.
## a vs b a vs c a vs d b vs c b vs d c vs d
## 1.025526 1.028896 -1.063345 1.003286 -1.090489 -1.094072
## a vs b a vs c a vs d b vs c b vs d c vs d
## 1.0255264 1.0288962 0.9404284 1.0032859 0.9170201 0.9140168
## a vs b a vs c a vs d b vs c b vs d c vs d
## 0.036364688 0.041097415 -0.088609974 0.004732726 -0.124974662 -0.129707388
The function returns a modified FC. That is, if the FC is smaller than 1 it is transformed to -1/FC. One can also use other functions than the mean for the aggregation of the data.
## a vs b a vs c a vs d b vs c b vs d c vs d
## 0.06301909 0.19666124 -0.00200599 0.13364215 -0.06502508 -0.19866723
Arbitrary Criteria
Furthermore, function pairwise.fun enables the application of arbitrary functions for pairwise comparisons.
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen
## keinen exakten p-Wert Berechnen
##
## Pairwise comparisons using Wilcoxon rank sum test with continuity correction
##
## data: airquality$Ozone and airquality$Month
##
## 5 6 7 8
## 6 0.19250 - - -
## 7 3e-05 0.01414 - -
## 8 0.00012 0.02591 0.86195 -
## 9 0.11859 0.95887 0.00074 0.00325
##
## P value adjustment method: none
## Package 'exactRankTests' is no longer under development.
## Please consider using package 'coin' instead.
## 5 vs 6 5 vs 7 5 vs 8 5 vs 9 6 vs 7 6 vs 8
## 1.897186e-01 1.087205e-05 6.108735e-05 1.171796e-01 1.183753e-02 2.333564e-02
## 6 vs 9 7 vs 8 7 vs 9 8 vs 9
## 9.528659e-01 8.595683e-01 5.281796e-04 2.714694e-03
Binary Classification
PPV and NPV
In case of medical diagnostic tests, usually sensitivity and specificity of the tests are known and there is also at least a rough estimate of the prevalence of the tested disease. In the practival application, the positive predictive value (PPV) and the negative predictive value are of crucial importance.
## PPV NPV
## 0.3333333 0.9999990
## PPV NPV
## 0.999992 0.999001
Performance Measures and Scores
In the development of diagnostic tests and more general in binary classification a variety of performance measures and scores can be found in literature. Functions perfMeasures and prefScores compute several of them.
## Measure Value
## 1 accuracy (ACC) 0.714
## 2 probabiliy of correct classification (PCC) 0.714
## 3 probability of missclassification (PMC) 0.286
## 4 error rate 0.286
## 5 sensitivity 0.337
## 6 specificity 0.903
## 7 prevalence 0.335
## 8 no information rate 0.665
## 9 weighted accuracy (wACC) 0.620
## 10 balanced accuracy (BACC) 0.620
## 11 informedness 0.240
## 12 Youden's J statistic 0.240
## 13 positive likelihood ratio (PLR) 3.479
## 14 negative likelihood ratio (NLR) 0.734
## 15 positive predictive value (PPV) 0.636
## 16 negative predictive value (NPV) 0.730
## 17 markedness 0.367
## 18 weighted predictive value 0.683
## 19 balanced predictive value 0.683
## 20 F1 score 0.441
## 21 Matthews' correlation coefficient (MCC) 0.297
## 22 proportion of positive predictions 0.177
## 23 expected accuracy 0.607
## 24 Cohen's kappa coefficient 0.272
## 25 detection rate 0.113
## Score Value
## 1 area under the ROC curve (AUC) 0.729
## 2 Gini index 0.457
## 3 Brier score 0.191
## 4 positive Brier score 0.361
## 5 negative Brier score 0.106
## 6 weighted Brier score 0.233
## 7 balanced Brier score 0.233
## Measure Value
## 1 accuracy (ACC) 0.714
## 2 probabiliy of correct classification (PCC) 0.714
## 3 probability of missclassification (PMC) 0.286
## 4 error rate 0.286
## 5 sensitivity 0.337
## 6 specificity 0.903
## 7 prevalence 0.335
## 8 no information rate 0.665
## 9 weighted accuracy (wACC) 0.620
## 10 balanced accuracy (BACC) 0.620
## 11 informedness 0.240
## 12 Youden's J statistic 0.240
## 13 positive likelihood ratio (PLR) 3.479
## 14 negative likelihood ratio (NLR) 0.734
## 15 positive predictive value (PPV) 0.636
## 16 negative predictive value (NPV) 0.730
## 17 markedness 0.367
## 18 weighted predictive value 0.683
## 19 balanced predictive value 0.683
## 20 F1 score 0.441
## 21 Matthews' correlation coefficient (MCC) 0.297
## 22 proportion of positive predictions 0.177
## 23 expected accuracy 0.607
## 24 Cohen's kappa coefficient 0.272
## 25 detection rate 0.113
## Score Value
## 1 area under the ROC curve (AUC) 0.729
## 2 Gini index 0.457
## 3 Brier score 0.191
## 4 positive Brier score 0.361
## 5 negative Brier score 0.106
## 6 weighted Brier score 0.233
## 7 balanced Brier score 0.233
## Measure Value
## 1 accuracy (ACC) 0.714
## 2 probabiliy of correct classification (PCC) 0.714
## 3 probability of missclassification (PMC) 0.286
## 4 error rate 0.286
## 5 sensitivity 0.337
## 6 specificity 0.903
## 7 prevalence 0.335
## 8 no information rate 0.665
## 9 weighted accuracy (wACC) 0.733
## 10 balanced accuracy (BACC) 0.620
## 11 informedness 0.240
## 12 Youden's J statistic 0.240
## 13 positive likelihood ratio (PLR) 3.479
## 14 negative likelihood ratio (NLR) 0.734
## 15 positive predictive value (PPV) 0.636
## 16 negative predictive value (NPV) 0.730
## 17 markedness 0.367
## 18 weighted predictive value 0.702
## 19 balanced predictive value 0.683
## 20 F1 score 0.441
## 21 Matthews' correlation coefficient (MCC) 0.297
## 22 proportion of positive predictions 0.177
## 23 expected accuracy 0.607
## 24 Cohen's kappa coefficient 0.272
## 25 detection rate 0.113
## Score Value
## 1 area under the ROC curve (AUC) 0.729
## 2 Gini index 0.457
## 3 Brier score 0.191
## 4 positive Brier score 0.361
## 5 negative Brier score 0.106
## 6 weighted Brier score 0.182
## 7 balanced Brier score 0.233
Optimal Cutoff
The function optCutoff computes the optimal cutoff for various performance weasures for binary classification. More precisely, all performance measures that are implemented in function perfMeasures.
## Optimal Cut-off Youden's J statistic
## 0.37504 0.34700
The computation of an optimal cut-off doesn’t make any sense for continuous scoring rules as their computation does not involve any cut-off (discretization/dichotomization).
Hosmer-Lemeshow and le Cessie-van Houwelingen-Copas-Hosmer
These tests are used to investigate the goodness of fit in logistic regression.
## Warning in HLgof.test(fit = pred, obs = infert$case): Found only 6 different
## groups for Hosmer-Lemesho C statistic.
## $C
##
## Hosmer-Lemeshow C statistic
##
## data: pred and infert$case
## X-squared = 4.4272, df = 4, p-value = 0.3513
##
##
## $H
##
## Hosmer-Lemeshow H statistic
##
## data: pred and infert$case
## X-squared = 6.3168, df = 8, p-value = 0.6118
## Warning in HLgof.test(fit = pred, obs = infert$case, X = model.matrix(case ~ :
## Found only 6 different groups for Hosmer-Lemesho C statistic.
## $C
##
## Hosmer-Lemeshow C statistic
##
## data: pred and infert$case
## X-squared = 4.4272, df = 4, p-value = 0.3513
##
##
## $H
##
## Hosmer-Lemeshow H statistic
##
## data: pred and infert$case
## X-squared = 6.3168, df = 8, p-value = 0.6118
##
##
## $gof
##
## le Cessie-van Houwelingen-Copas-Hosmer global goodness of fit test
##
## data: pred and infert$case
## z = 1.7533, p-value = 0.07954
Sample Size Calculation
Given an expected sensitivity and specificity we can compute sample size, power, delta or significance level of diagnostic test.
##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 40
## n1 = 40
## delta = 0.14
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of cases, n1 is number of controls
##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 43
## n1 = 43
## delta = 0.13
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of cases, n1 is number of controls
##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 47
## n1 = 47
## delta = 0.12
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of cases, n1 is number of controls
The sample size planning for developing binary classifiers in case of high dimensional data, we can apply function ssize.pcc, which is based on the probability of correct classification (PCC).
##
## Sample Size Planning for Developing Classifiers Using High Dimensional Data
##
## gamma = 0.1
## prev = 0.5
## nrFeatures = 22000
## n1 = 21
## n2 = 21
##
## NOTE: n1 is number of cases, n2 is number of controls
Omics Data
Aggregating Technical Replicates
In case of omics experiments it is often the case that technical replicates are determined and hence it is part of the preprocessing of the raw data to aggregate these technical replicates. This is the purpose of function repMeans.
## $exprs
## [,1] [,2] [,3] [,4] [,5]
## [1,] -1.1561674 1.6659909 -0.7913139 -1.9251453 -0.3599751
## [2,] 0.1467085 -1.1392006 0.4605913 0.4127695 0.1887531
## [3,] -0.0974125 1.5408878 0.5620414 -1.1852888 -0.8445834
## [4,] 0.9112097 -0.5272343 -0.3865341 0.3013137 -0.5722881
##
## $flags
## [,1] [,2] [,3] [,4] [,5]
## [1,] 11 11 18 13 14
## [2,] 13 13 9 17 15
## [3,] 13 12 12 14 13
## [4,] 17 8 18 15 13
1- and 2-Way ANOVA
Functions oneWayAnova and twoWayAnova return a function that can be used to perform a 1- or 2-way ANOVA, respectively.
## [1] 0.4775058
x <- matrix(rnorm(12*10), nrow = 10)
## 2-way ANOVA with interaction
af1 <- twoWayAnova(c(rep(1,6),rep(2,6)), rep(c(rep(1,3), rep(2,3)), 2))
## p values
apply(x, 1, af1)
## [,1] [,2] [,3] [,4] [,5] [,6]
## cov1 0.8600392 0.3933937 0.3637042 0.25648878 0.6445050 0.3662116
## cov2 0.8735542 0.2238443 0.8970489 0.04185771 0.8627222 0.7268275
## interaction 0.7134877 0.4613755 0.3382075 0.95756365 0.6680431 0.9783548
## [,7] [,8] [,9] [,10]
## cov1 0.07973061 0.5322539 0.5296932 0.1703228
## cov2 0.01711811 0.2941207 0.2012428 0.9128464
## interaction 0.01450295 0.4349538 0.3198896 0.9204181
## 2-way ANOVA without interaction
af2 <- twoWayAnova(c(rep(1,6),rep(2,6)), rep(c(rep(1,3), rep(2,3)), 2),
interaction = FALSE)
## p values
apply(x, 1, af2)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## cov1 0.8524495 0.3801684 0.3616139 0.22726078 0.6275373 0.3362318 0.18581834
## cov2 0.8666834 0.2103309 0.8969023 0.03037705 0.8557404 0.7100415 0.06095909
## [,8] [,9] [,10]
## cov1 0.5228496 0.5304601 0.1447322
## cov2 0.2823267 0.2000102 0.9073261
Correlation Distance Matrix
In the analysis of omics data correlation and absolute correlation distance matrices are often used during quality control. Function corDist can compute the Pearson, Spearman, Kendall or Cosine sample correlation and absolute correlation as well as the minimum covariance determinant or the orthogonalized Gnanadesikan-Kettenring correlation and absolute correlation.
## 1 2 3 4
## 2 0.9958159
## 3 1.0054874 1.0293796
## 4 1.0700867 0.9880482 1.0138094
## 5 0.9996421 0.9906085 1.0014355 0.4217543
## 1 2 3 4
## 2 1.0799925
## 3 0.9866692 1.0113658
## 4 0.8290512 1.0834606 1.1585555
## 5 0.9807935 1.0018795 0.8963924 1.0837111
## 1 2 3 4
## 2 0.9975890
## 3 1.0935537 0.9548022
## 4 0.8083605 1.0899254 1.2154796
## 5 0.8453323 0.9413417 1.0646901 1.0704574
MAD Matrix
In case of outliers the MAD may be useful as dispersion measure.
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.000000 3.625866 3.226823 2.826532 2.876544
## [2,] 3.625866 0.000000 3.213304 2.626208 2.768792
## [3,] 3.226823 3.213304 0.000000 3.013375 2.721847
## [4,] 2.826532 2.626208 3.013375 0.000000 2.824410
## [5,] 2.876544 2.768792 2.721847 2.824410 0.000000
Similarity Matrices
First, we can plot a similarity matrix based on correlation.
Next, we can use the MAD.
Colors for Heatmaps
Nowadays there are better solutions e.g. provided by Bioconductor package complexHeatmaps. This was my solution to get a better coloring of heatmaps.
## generate some random data
data.plot <- matrix(rnorm(100*50, sd = 1), ncol = 50)
colnames(data.plot) <- paste("patient", 1:50)
rownames(data.plot) <- paste("gene", 1:100)
data.plot[1:70, 1:30] <- data.plot[1:70, 1:30] + 3
data.plot[71:100, 31:50] <- data.plot[71:100, 31:50] - 1.4
data.plot[1:70, 31:50] <- rnorm(1400, sd = 1.2)
data.plot[71:100, 1:30] <- rnorm(900, sd = 1.2)
nrcol <- 128
## Load required packages
library(gplots)
##
## Attache Paket: 'gplots'
## Das folgende Objekt ist maskiert 'package:stats':
##
## lowess
String Alignment
String Distances
In Bioinformatics the (pairwise and multiple) alignment of strings is an important topic. For this one can use the distance or similarity between strings. The Hamming and the the Levenshtein (edit) distance are implemented in function stringDist.
## GACGGATTATG
## GATCGGAATAG 3
## GACGGATTATG
## GATCGGAATAG 3
In case of the Levenshtein (edit) distance, the respective scoring and traceback matrices are attached as attributes to the result.
## gap G A T C G G A A T A G
## gap 0 1 2 3 4 5 6 7 8 9 10 11
## G 1 0 1 2 3 4 5 6 7 8 9 10
## A 2 1 0 1 2 3 4 5 6 7 8 9
## C 3 2 1 1 1 2 3 4 5 6 7 8
## G 4 3 2 2 2 1 2 3 4 5 6 7
## G 5 4 3 3 3 2 1 2 3 4 5 6
## A 6 5 4 4 4 3 2 1 2 3 4 5
## T 7 6 5 4 5 4 3 2 2 2 3 4
## T 8 7 6 5 5 5 4 3 3 2 3 4
## A 9 8 7 6 6 6 5 4 3 3 2 3
## T 10 9 8 7 7 7 6 5 4 3 3 3
## G 11 10 9 8 8 7 7 6 5 4 4 3
## gap G A T C G G A A T A
## gap "start" "i" "i" "i" "i" "i" "i" "i" "i" "i" "i"
## G "d" "m" "i" "i" "i" "m/i" "m/i" "i" "i" "i" "i"
## A "d" "d" "m" "i" "i" "i" "i" "m/i" "m/i" "i" "m/i"
## C "d" "d" "d" "mm" "m" "i" "i" "i" "i" "i" "i"
## G "d" "d/m" "d" "d/mm" "d/mm" "m" "m/i" "i" "i" "i" "i"
## G "d" "d/m" "d" "d/mm" "d/mm" "d/m" "m" "i" "i" "i" "i"
## A "d" "d" "d/m" "d/mm" "d/mm" "d" "d" "m" "m/i" "i" "m/i"
## T "d" "d" "d" "m" "d/mm/i" "d" "d" "d" "mm" "m" "i"
## T "d" "d" "d" "d/m" "mm" "d" "d" "d" "d/mm" "m" "mm/i"
## A "d" "d" "d/m" "d" "d/mm" "d/mm" "d" "d/m" "m" "d" "m"
## T "d" "d" "d" "d/m" "d/mm" "d/mm" "d" "d" "d" "m" "d"
## G "d" "d/m" "d" "d" "d/mm" "m" "d/m" "d" "d" "d" "d/mm"
## G
## gap "i"
## G "m/i"
## A "i"
## C "i"
## G "m/i"
## G "m/i"
## A "i"
## T "i"
## T "mm/i"
## A "i"
## T "mm"
## G "m"
The characters in the trace-back matrix reflect insertion of a gap in string y (d: deletion), match (m), mismatch (mm), and insertion of a gap in string x (i).
String Similarities
The function stringSim computes the optimal alignment scores for global (Needleman-Wunsch) and local (Smith-Waterman) alignments with constant gap penalties. Scoring and trace-back matrix are again attached as attributes to the results.
## GACGGATTATG
## GATCGGAATAG 6
## gap G A T C G G A A T A G
## gap 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11
## G -1 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9
## A -2 0 2 1 0 -1 -2 -3 -4 -5 -6 -7
## C -3 -1 1 1 2 1 0 -1 -2 -3 -4 -5
## G -4 -2 0 0 1 3 2 1 0 -1 -2 -3
## G -5 -3 -1 -1 0 2 4 3 2 1 0 -1
## A -6 -4 -2 -2 -1 1 3 5 4 3 2 1
## T -7 -5 -3 -1 -2 0 2 4 4 5 4 3
## T -8 -6 -4 -2 -2 -1 1 3 3 5 4 3
## A -9 -7 -5 -3 -3 -2 0 2 4 4 6 5
## T -10 -8 -6 -4 -4 -3 -1 1 3 5 5 5
## G -11 -9 -7 -5 -5 -3 -2 0 2 4 4 6
## gap G A T C G G A A T A
## gap "start" "i" "i" "i" "i" "i" "i" "i" "i" "i" "i"
## G "d" "m" "i" "i" "i" "m/i" "m/i" "i" "i" "i" "i"
## A "d" "d" "m" "i" "i" "i" "i" "m/i" "m/i" "i" "m/i"
## C "d" "d" "d" "mm" "m" "i" "i" "i" "i" "i" "i"
## G "d" "d/m" "d" "d/mm" "d" "m" "m/i" "i" "i" "i" "i"
## G "d" "d/m" "d" "d/mm" "d" "d/m" "m" "i" "i" "i" "i"
## A "d" "d" "d/m" "d/mm" "d" "d" "d" "m" "m/i" "i" "m/i"
## T "d" "d" "d" "m" "d/i" "d" "d" "d" "mm" "m" "i"
## T "d" "d" "d" "d/m" "mm" "d" "d" "d" "d/mm" "m" "mm/i"
## A "d" "d" "d/m" "d" "d/mm" "d" "d" "d/m" "m" "d" "m"
## T "d" "d" "d" "d/m" "d/mm" "d" "d" "d" "d" "m" "d"
## G "d" "d/m" "d" "d" "d/mm" "m" "d/m" "d" "d" "d" "d/mm"
## G
## gap "i"
## G "m/i"
## A "i"
## C "i"
## G "m/i"
## G "m/i"
## A "i"
## T "i"
## T "mm/i"
## A "i"
## T "mm"
## G "m"
## GACGGATTATG
## GATCGGAATAG 6
## gap G A T C G G A A T A G
## gap 0 0 0 0 0 0 0 0 0 0 0 0
## G 0 1 0 0 0 1 1 0 0 0 0 1
## A 0 0 2 1 0 0 0 2 1 0 1 0
## C 0 0 1 1 2 1 0 1 1 0 0 0
## G 0 1 0 0 1 3 2 1 0 0 0 1
## G 0 1 0 0 0 2 4 3 2 1 0 1
## A 0 0 2 1 0 1 3 5 4 3 2 1
## T 0 0 1 3 2 1 2 4 4 5 4 3
## T 0 0 0 2 2 1 1 3 3 5 4 3
## A 0 0 1 1 1 1 0 2 4 4 6 5
## T 0 0 0 2 1 0 0 1 3 5 5 5
## G 0 1 0 1 1 2 1 0 2 4 4 6
## gap G A T C G
## gap "start" "i" "i" "i" "i" "i"
## G "d" "m" "i/stop" "stop" "stop" "m"
## A "d" "d/stop" "m" "i" "i/stop" "d/stop"
## C "d" "stop" "d" "mm" "m" "i"
## G "d" "m" "d/i/stop" "d/mm/stop" "d" "m"
## G "d" "m" "mm/i/stop" "stop" "d/stop" "d/m"
## A "d" "d/stop" "m" "i" "i/stop" "d"
## T "d" "stop" "d" "m" "i" "i"
## T "d" "stop" "d/stop" "d/m" "mm" "mm/i"
## A "d" "stop" "m" "d" "d/mm" "mm"
## T "d" "stop" "d/stop" "m" "i" "d/mm/i/stop"
## G "d" "m" "i/stop" "d" "mm" "m"
## G A A T A G
## gap "i" "i" "i" "i" "i" "i"
## G "m" "i/stop" "stop" "stop" "stop" "m"
## A "d/mm/stop" "m" "m/i" "i/stop" "m" "d/i/stop"
## C "i/stop" "d" "mm" "mm/i/stop" "d/stop" "mm/stop"
## G "m/i" "i" "d/mm/i/stop" "mm/stop" "stop" "m"
## G "m" "i" "i" "i" "i/stop" "m"
## A "d" "m" "m/i" "i" "m/i" "i"
## T "d" "d" "mm" "m" "i" "i"
## T "d" "d" "d/mm" "m" "mm/i" "mm/i"
## A "d/mm/i/stop" "d/m" "m" "d" "m" "i"
## T "mm/stop" "d" "d" "m" "d" "mm"
## G "m/i" "d/i/stop" "d" "d" "d/mm" "m"
The entry stop indicates that the minimum similarity score has been reached.
Optimal Alignment
Finally, the function traceBack computes an optimal global or local alignment based on a trace back matrix as provided by function stringDist or stringSim.
## pairwise global alignment
## x' "GA-CGGATTATG"
## y' "GATCGGAATA-G"
## pairwise global alignment
## x' "GA-CGGATTATG"
## y' "GATCGGAATA-G"
## pairwise local alignment
## x' "GA-CGGATTA"
## y' "GATCGGAATA"
