Revision as of 22:34, 7 September 2019

Statisticians

Karl Pearson (1857-1936): chi-square, p-value, PCA
William Sealy Gosset (1876-1937): Student's t
Ronald Fisher (1890-1962): ANOVA
Egon Pearson (1895-1980): son of Karl Pearson
Jerzy Neyman (1894-1981): type 1 error

Statistics for biologists

http://www.nature.com/collections/qghhqm

Transform sample values to their percentiles

https://stackoverflow.com/questions/21219447/calculating-percentile-of-dataset-column

set.seed(1234)
x <- rnorm(10)
x
# [1] -1.2070657  0.2774292  1.0844412 -2.3456977  0.4291247  0.5060559
# [7] -0.5747400 -0.5466319 -0.5644520 -0.8900378
ecdf(x)(x)
# [1] 0.2 0.7 1.0 0.1 0.8 0.9 0.4 0.6 0.5 0.3

rank(x)
# [1]  2  7 10  1  8  9  4  6  5  3

Box(Box and whisker) plot in R

See

https://en.wikipedia.org/wiki/Box_plot
https://owi.usgs.gov/blog/boxplots/ (ggplot2 is used)
https://flowingdata.com/2008/02/15/how-to-read-and-use-a-box-and-whisker-plot/
Quartile from Wikipedia. The quartiles returned from R are the same as the method defined by Method 2 described in Wikipedia.

An example for a graphical explanation.

> x=c(0,4,15, 1, 6, 3, 20, 5, 8, 1, 3)
> summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      0       2       4       6       7      20 
> sort(x)
 [1]  0  1  1  3  3  4  5  6  8 15 20
> boxplot(x, col = 'grey')

# https://en.wikipedia.org/wiki/Quartile#Example_1
> summary(c(6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   6.00   25.50   40.00   33.18   42.50   49.00

File:Boxplot.svg

The lower and upper edges of box is determined by the first and 3rd quartiles (2 and 7 in the above example).
- 2 = median(c(0, 1, 1, 3, 3, 4)) = (1+3)/2
- 7 = median(c(4, 5, 6, 8, 15, 20)) = (6+8)/2
- IQR = 7 - 2 = 5
The thick dark horizon line is the median (4 in the example).
Outliers are defined by (the empty circles in the plot)
- Observations larger than 3rd quartile + 1.5 * IQR (7+1.5*5=14.5) and
- smaller than 1st quartile - 1.5 * IQR (2-1.5*5=-5.5).
- Note that the cutoffs are not shown in the Box plot.
Whisker (defined using the cutoffs used to define outliers)
- Upper whisker is defined by the largest "data" below 3rd quartile + 1.5 * IQR (8 in this example), and
- Lower whisker is defined by the smallest "data" greater than 1st quartile - 1.5 * IQR (0 in this example).
- See another example below where we can see the whiskers fall on observations.

Note the wikipedia lists several possible definitions of a whisker. R uses the 2nd method (Tukey boxplot) to define whiskers.

Create boxplots from a list object

Normally we use a vector to create a single boxplot or a formula on a data to create boxplots.

But we can also use split() to create a list and then make boxplots.

Dot-box plot

http://civilstat.com/2012/09/the-grammar-of-graphics-notes-on-first-reading/
http://www.r-graph-gallery.com/89-box-and-scatter-plot-with-ggplot2/
http://www.sthda.com/english/wiki/ggplot2-box-plot-quick-start-guide-r-software-and-data-visualization

Graphs in R – Overlaying Data Summaries in Dotplots. Note that for some reason, the boxplot will cover the dots when we save the plot to an svg or a png file. So an alternative solution is to change the order

par(cex.main=0.9,cex.lab=0.8,font.lab=2,cex.axis=0.8,font.axis=2,col.axis="grey50")
boxplot(weight ~ feed, data = chickwts, range=0, whisklty = 0, staplelty = 0)
par(new = TRUE)
stripchart(weight ~ feed, data = chickwts, xlim=c(0.5,6.5), vertical=TRUE, method="stack", offset=0.8, pch=19,
 main = "Chicken weights after six weeks", xlab = "Feed Type", ylab = "Weight (g)")

File:Boxdot.svg

Other boxplots

File:Lotsboxplot.png

stem and leaf plot

stem(). See R Tutorial.

Note that stem plot is useful when there are outliers.

> stem(x)

  The decimal point is 10 digit(s) to the right of the |

   0 | 00000000000000000000000000000000000000000000000000000000000000000000+419
   1 |
   2 |
   3 |
   4 |
   5 |
   6 |
   7 |
   8 |
   9 |
  10 |
  11 |
  12 | 9

> max(x)
[1] 129243100275
> max(x)/1e10
[1] 12.92431

> stem(y)

  The decimal point is at the |

  0 | 014478
  1 | 0
  2 | 1
  3 | 9
  4 | 8

> y
 [1] 3.8667356428 0.0001762708 0.7993462430 0.4181079732 0.9541728562
 [6] 4.7791262101 0.6899313108 2.1381289177 0.0541736818 0.3868776083

> set.seed(1234)
> z <- rnorm(10)*10
> z
 [1] -12.070657   2.774292  10.844412 -23.456977   4.291247   5.060559
 [7]  -5.747400  -5.466319  -5.644520  -8.900378
> stem(z)

  The decimal point is 1 digit(s) to the right of the |

  -2 | 3
  -1 | 2
  -0 | 9665
   0 | 345
   1 | 1

Box-Cox transformation

the Holy Trinity (LRT, Wald, Score tests)

https://en.wikipedia.org/wiki/Likelihood_function which includes profile likelihood and partial likelihood
Review of the likelihood theory
The “Three Plus One” Likelihood-Based Test Statistics: Unified Geometrical and Graphical Interpretations
Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.
- Score test is step-up. Score test is typically used in forward steps to screen covariates currently not included in a model for their ability to improve model.
- Wald test is step-down. Wald test starts at the full model. It evaluate the significance of a variable by comparing the ratio of its estimate and its standard error with an appropriate t distribution (for linear models) or standard normal distribution (for logistic or Cox regression).
- Likelihood ratio tests provide the best control over nuisance parameters by maximizing the likelihood over them both in H0 model and H1 model. In particular, if several coefficients are being tested simultaneously, LRTs for model comparison are preferred over Wald or score tests.
R packages
- lmtest package, waldtest() and lrtest().

Don't invert that matrix

Different matrix decompositions/factorizations

QR decomposition, qr()
LU decomposition, lu() from the 'Matrix' package
Cholesky decomposition, chol()
Singular value decomposition, svd()

set.seed(1234)
x <- matrix(rnorm(10*2), nr= 10)
cmat <- cov(x); cmat
# [,1]       [,2]
# [1,]  0.9915928 -0.1862983
# [2,] -0.1862983  1.1392095

# cholesky decom
d1 <- chol(cmat)
t(d1) %*% d1  # equal to cmat
d1  # upper triangle
# [,1]       [,2]
# [1,] 0.9957875 -0.1870864
# [2,] 0.0000000  1.0508131

# svd
d2 <- svd(cmat)
d2$u %*% diag(d2$d) %*% t(d2$v) # equal to cmat
d2$u %*% diag(sqrt(d2$d))
# [,1]      [,2]
# [1,] -0.6322816 0.7692937
# [2,]  0.9305953 0.5226872

Linear Regression

Regression Models for Data Science in R by Brian Caffo

Comic https://xkcd.com/1725/

Different models (in R)

http://www.quantide.com/raccoon-ch-1-introduction-to-linear-models-with-r/

dummy.coef.lm() in R

Extracts coefficients in terms of the original levels of the coefficients rather than the coded variables.

model.matrix, design matrix

ExploreModelMatrix: Explore design matrices interactively with R/Shiny

Contrasts in linear regression

Page 147 of Modern Applied Statistics with S (4th ed)
https://biologyforfun.wordpress.com/2015/01/13/using-and-interpreting-different-contrasts-in-linear-models-in-r/ This explains the meanings of 'treatment', 'helmert' and 'sum' contrasts.

A (sort of) Complete Guide to Contrasts in R by Rose Maier

mat

##      constant NLvMH  NvL  MvH
## [1,]        1  -0.5  0.5  0.0
## [2,]        1  -0.5 -0.5  0.0
## [3,]        1   0.5  0.0  0.5
## [4,]        1   0.5  0.0 -0.5
mat <- mat[ , -1]

model7 <- lm(y ~ dose, data=data, contrasts=list(dose=mat) )
summary(model7)

## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  <br />
## (Intercept)  118.578      1.076 110.187  < 2e-16 ***
## doseNLvMH      3.179      2.152   1.477  0.14215  <br />
## doseNvL       -8.723      3.044  -2.866  0.00489 ** 
## doseMvH       13.232      3.044   4.347 2.84e-05 ***

# double check your contrasts
attributes(model7$qr$qr)$contrasts
## $dose
##      NLvMH  NvL  MvH
## None  -0.5  0.5  0.0
## Low   -0.5 -0.5  0.0
## Med    0.5  0.0  0.5
## High   0.5  0.0 -0.5

library(dplyr)
dose.means <- summarize(group_by(data, dose), y.mean=mean(y))
dose.means
## Source: local data frame [4 x 2]
## 
##   dose   y.mean
## 1 None 112.6267
## 2  Low 121.3500
## 3  Med 126.7839
## 4 High 113.5517

# The coefficient estimate for the first contrast (3.18) equals the average of 
# the last two groups (126.78 + 113.55 /2 = 120.17) minus the average of 
# the first two groups (112.63 + 121.35 /2 = 116.99).

Multicollinearity

Multicollinearity in R
alias: Find Aliases (Dependencies) In A Model

> op <- options(contrasts = c("contr.helmert", "contr.poly"))
> npk.aov <- aov(yield ~ block + N*P*K, npk)
> alias(npk.aov)
Model :
yield ~ block + N * P * K

Complete :
         (Intercept) block1 block2 block3 block4 block5 N1    P1    K1    N1:P1 N1:K1 P1:K1
N1:P1:K1     0           1    1/3    1/6  -3/10   -1/5      0     0     0     0     0     0

> options(op)

Exposure

https://en.mimi.hu/mathematics/exposure_variable.html

Independent variable = predictor = explanatory = exposure variable

Confounders, confounding

Causal inference

Confidence interval vs prediction interval

Confidence intervals tell you about how well you have determined the mean E(Y). Prediction intervals tell you where you can expect to see the next data point sampled. That is, CI is computed using Var(E(Y|X)) and PI is computed using Var(E(Y|X) + e).

Heteroskedasticity

Dealing with heteroskedasticity; regression with robust standard errors using R

Linear regression with Map Reduce

https://freakonometrics.hypotheses.org/53269

Relationship between multiple variables

Visualizing the relationship between multiple variables

Quantile regression

Non- and semi-parametric regression

Mean squared error

Splines

https://en.wikipedia.org/wiki/B-spline
Cubic and Smoothing Splines in R. bs() is for cubic spline and smooth.spline() is for smoothing spline.
Can we use B-splines to generate non-linear data?
How to force passing two data points? (cobs package)
https://www.rdocumentation.org/packages/cobs/versions/1.3-3/topics/cobs

k-Nearest neighbor regression

k-NN regression in practice: boundary problem, discontinuities problem.
Weighted k-NN regression: want weight to be small when distance is large. Common choices - weight = kernel(xi, x)

Kernel regression

Instead of weighting NN, weight ALL points. Nadaraya-Watson kernel weighted average:

\(\displaystyle \hat{y}_q = \sum c_{qi} y_i/\sum c_{qi} = \frac{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))*y_i}{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))} \).

Choice of bandwidth \(\displaystyle \lambda\) for bias, variance trade-off. Small \(\displaystyle \lambda\) is over-fitting. Large \(\displaystyle \lambda\) can get an over-smoothed fit. Cross-validation.
Kernel regression leads to locally constant fit.
Issues with high dimensions, data scarcity and computational complexity.

Principal component analysis

See PCA.

Partial Least Squares (PLS)

https://en.wikipedia.org/wiki/Partial_least_squares_regression. The general underlying model of multivariate PLS is

\(\displaystyle X = T P^\mathrm{T} + E\)

\(\displaystyle Y = U Q^\mathrm{T} + F\)

where Template:Mvar is an \(\displaystyle n \times m\) matrix of predictors, Template:Mvar is an \(\displaystyle n \times p\) matrix of responses; Template:Mvar and Template:Mvar are \(\displaystyle n \times l\) matrices that are, respectively, projections of Template:Mvar (the X score, component or factor matrix) and projections of Template:Mvar (the Y scores); Template:Mvar and Template:Mvar are, respectively, \(\displaystyle m \times l\) and \(\displaystyle p \times l\) orthogonal loading matrices; and matrices Template:Mvar and Template:Mvar are the error terms, assumed to be independent and identically distributed random normal variables. The decompositions of Template:Mvar and Template:Mvar are made so as to maximise the covariance between Template:Mvar and Template:Mvar (projection matrices).

Supervised vs. Unsupervised Learning: Exploring Brexit with PLS and PCA
pls R package
plsRcox R package (archived). See here for the installation.

PLS, PCR (principal components regression) and ridge regression tend to behave similarly. Ridge regression may be preferred because it shrinks smoothly, rather than in discrete steps.

High dimension

Partial least squares prediction in high-dimensional regression Cook and Forzani, 2019

Feature selection

https://en.wikipedia.org/wiki/Feature_selection

Independent component analysis

ICA is another dimensionality reduction method.

ICA vs PCA

ICS vs FA

Correspondence analysis

https://francoishusson.wordpress.com/2017/07/18/multiple-correspondence-analysis-with-factominer/ and the book Exploratory Multivariate Analysis by Example Using R

t-SNE

t-Distributed Stochastic Neighbor Embedding (t-SNE) is a technique for dimensionality reduction that is particularly well suited for the visualization of high-dimensional datasets.

Visualize the random effects

http://www.quantumforest.com/2012/11/more-sense-of-random-effects/

Calibration

How to determine calibration accuracy/uncertainty of a linear regression?
Linear Regression and Calibration Curves
Regression and calibration Shaun Burke
calibrate package

The index of prediction accuracy: an intuitive measure useful for evaluating risk prediction models by Kattan and Gerds 2018. The following code demonstrates Figure 2.

# Odds ratio =1 and calibrated model
set.seed(666)
x = rnorm(1000)         <br />
z1 = 1 + 0*x      <br />
pr1 = 1/(1+exp(-z1))
y1 = rbinom(1000,1,pr1)<br />
mean(y1) # .724, marginal prevalence of the outcome
dat1 <- data.frame(x=x, y=y1)
newdat1 <- data.frame(x=rnorm(1000), y=rbinom(1000, 1, pr1))

# Odds ratio =1 and severely miscalibrated model
set.seed(666)
x = rnorm(1000)         <br />
z2 =  -2 + 0*x      <br />
pr2 = 1/(1+exp(-z2))<br />
y2 = rbinom(1000,1,pr2)<br />
mean(y2) # .12
dat2 <- data.frame(x=x, y=y2)
newdat2 <- data.frame(x=rnorm(1000), y=rbinom(1000, 1, pr2))

library(riskRegression)
lrfit1 <- glm(y ~ x, data = dat1, family = 'binomial')
IPA(lrfit1, newdata = newdat1)
#     Variable     Brier           IPA     IPA.gain
# 1 Null model 0.1984710  0.000000e+00 -0.003160010
# 2 Full model 0.1990982 -3.160010e-03  0.000000000
# 3          x 0.1984800 -4.534668e-05 -0.003114664
1 - 0.1990982/0.1984710
# [1] -0.003160159

lrfit2 <- glm(y ~ x, family = 'binomial')
IPA(lrfit2, newdata = newdat1)
#     Variable     Brier       IPA     IPA.gain
# 1 Null model 0.1984710  0.000000 -1.859333763
# 2 Full model 0.5674948 -1.859334  0.000000000
# 3          x 0.5669200 -1.856437 -0.002896299
1 - 0.5674948/0.1984710
# [1] -1.859334

From the simulated data, we see IPA = -3.16e-3 for a calibrated model and IPA = -1.86 for a severely miscalibrated model.

ROC curve

See ROC.

NRI (Net reclassification improvement)

Maximum likelihood

Difference of partial likelihood, profile likelihood and marginal likelihood

Generalized Linear Model

Lectures from a course in Simon Fraser University Statistics.

Doing magic and analyzing seasonal time series with GAM (Generalized Additive Model) in R

Link function

Link Functions versus Data Transforms

Quasi Likelihood

Quasi-likelihood is like log-likelihood. The quasi-score function (first derivative of quasi-likelihood function) is the estimating equation.

Original paper by Peter McCullagh.
Lecture 20 from SFU.
U. Washington and another lecture focuses on overdispersion.
This lecture contains a table of quasi likelihood from common distributions.

Plot

https://strengejacke.wordpress.com/2015/02/05/sjplot-package-and-related-online-manuals-updated-rstats-ggplot/

Deviance, stats::deviance() and glmnet::deviance.glmnet() from R

It is a generalization of the idea of using the sum of squares of residuals (RSS) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood. See What is Deviance? (specifically in CART/rpart) to manually compute deviance and compare it with the returned value of the deviance() function from a linear regression. Summary: deviance() = RSS in linear models.
https://www.rdocumentation.org/packages/stats/versions/3.4.3/topics/deviance
Likelihood ratio tests and the deviance http://data.princeton.edu/wws509/notes/a2.pdf#page=6
Deviance(y,muhat) = 2*(loglik_saturated - loglik_proposed)
Interpreting Residual and Null Deviance in GLM R
- Null Deviance = 2(LL(Saturated Model) - LL(Null Model)) on df = df_Sat - df_Null. The null deviance shows how well the response variable is predicted by a model that includes only the intercept (grand mean).
- Residual Deviance = 2(LL(Saturated Model) - LL(Proposed Model)) = \(\displaystyle 2(LL(y|y) - LL(\hat{\mu}|y))\), df = df_Sat - df_Proposed=n-p. ==> deviance() has returned.
- Null deviance > Residual deviance. Null deviance df = n-1. Residual deviance df = n-p.

## an example with offsets from Venables & Ripley (2002, p.189)
utils::data(anorexia, package = "MASS")

anorex.1 <- glm(Postwt ~ Prewt + Treat + offset(Prewt),
                family = gaussian, data = anorexia)
summary(anorex.1)

# Call:
#   glm(formula = Postwt ~ Prewt + Treat + offset(Prewt), family = gaussian, 
#       data = anorexia)
# 
# Deviance Residuals: 
#   Min        1Q    Median        3Q       Max<br />
# -14.1083   -4.2773   -0.5484    5.4838   15.2922<br />
# 
# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)  <br />
# (Intercept)  49.7711    13.3910   3.717 0.000410 ***
#   Prewt        -0.5655     0.1612  -3.509 0.000803 ***
#   TreatCont    -4.0971     1.8935  -2.164 0.033999 *<br />
#   TreatFT       4.5631     2.1333   2.139 0.036035 *<br />
#   ---
#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# (Dispersion parameter for gaussian family taken to be 48.69504)
# 
# Null deviance: 4525.4  on 71  degrees of freedom
# Residual deviance: 3311.3  on 68  degrees of freedom
# AIC: 489.97
# 
# Number of Fisher Scoring iterations: 2

deviance(anorex.1)
# [1] 3311.263

In glmnet package. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). Null deviance is defined to be 2*(loglike_sat -loglike(Null)); The NULL model refers to the intercept model, except for the Cox, where it is the 0 model. Hence dev.ratio=1-deviance/nulldev, and this deviance method returns (1-dev.ratio)*nulldev.
- What deviance is glmnet using to compare values of λ?

x=matrix(rnorm(100*2),100,2)
y=rnorm(100)
fit1=glmnet(x,y) 
deviance(fit1)  # one for each lambda
#  [1] 98.83277 98.53893 98.29499 98.09246 97.92432 97.78472 97.66883
#  [8] 97.57261 97.49273 97.41327 97.29855 97.20332 97.12425 97.05861
# ...
# [57] 96.73772 96.73770
fit2 <- glmnet(x, y, lambda=.1) # fix lambda
deviance(fit2)
# [1] 98.10212
deviance(glm(y ~ x))
# [1] 96.73762
sum(residuals(glm(y ~ x))^2)
# [1] 96.73762

Saturated model

The saturated model always has n parameters where n is the sample size.
Logistic Regression : How to obtain a saturated model

Simulate data

Density plot

# plot a Weibull distribution with shape and scale
func <- function(x) dweibull(x, shape = 1, scale = 3.38)
curve(func, .1, 10)

func <- function(x) dweibull(x, shape = 1.1, scale = 3.38)
curve(func, .1, 10)

The shape parameter plays a role on the shape of the density function and the failure rate.

Shape <=1: density is convex, not a hat shape.
Shape =1: failure rate (hazard function) is constant. Exponential distribution.
Shape >1: failure rate increases with time

Simulate data from a specified density

http://stackoverflow.com/questions/16134786/simulate-data-from-non-standard-density-function

Signal to noise ratio

\(\displaystyle \frac{\sigma^2_{signal}}{\sigma^2_{noise}} = \frac{Var(f(X))}{Var(e)} \) if Y = f(X) + e

Page 401 of ESLII (https://web.stanford.edu/~hastie/ElemStatLearn//) 12th print.

Some examples of signal to noise ratio

ESLII_print12.pdf: .64, 5, 4
Yuan and Lin 2006: 1.8, 3
A framework for estimating and testing qualitative interactions with applications to predictive biomarkers Roth, Biostatistics, 2018

Effect size, Cohen's d and volcano plot

https://en.wikipedia.org/wiki/Effect_size (See also the estimation by the pooled sd)

\(\displaystyle \theta = \frac{\mu_1 - \mu_2} \sigma,\)

Effect size, sample size and power from Learning statistics with R: A tutorial for psychology students and other beginners.
t-statistic and Cohen's d for the case of mean difference between two independent groups
Cohen’s D for Experimental Planning
Volcano plot
- Y-axis: -log(p)
- X-axis: log2 fold change OR effect size (Cohen's D). An example from RNA-Seq data.

Multiple comparisons

If you perform experiments over and over, you's bound to find something. So significance level must be adjusted down when performing multiple hypothesis tests.
http://www.gs.washington.edu/academics/courses/akey/56008/lecture/lecture10.pdf
Book 'Multiple Comparison Using R' by Bretz, Hothorn and Westfall, 2011.
Plot a histogram of p-values, a post from varianceexplained.org. The anti-conservative histogram (tail on the RHS) is what we have typically seen in e.g. microarray gene expression data.
Comparison of different ways of multiple-comparison in R.

Take an example, Suppose 550 out of 10,000 genes are significant at .05 level

P-value < .05 ==> Expect .05*10,000=500 false positives
False discovery rate < .05 ==> Expect .05*550 =27.5 false positives
Family wise error rate < .05 ==> The probablity of at least 1 false positive <.05

According to Lifetime Risk of Developing or Dying From Cancer, there is a 39.7% risk of developing a cancer for male during his lifetime (in other words, 1 out of every 2.52 men in US will develop some kind of cancer during his lifetime) and 37.6% for female. So the probability of getting at least one cancer patient in a 3-generation family is 1-.6**3 - .63**3 = 0.95.

False Discovery Rate

https://en.wikipedia.org/wiki/False_discovery_rate
Paper Definition by Benjamini and Hochberg in JRSS B 1995.
A comic
P-value vs false discovery rate vs family wise error rate. See 10 statistics tip or Statistics for Genomics (140.688) from Jeff Leek. Suppose 550 out of 10,000 genes are significant at .05 level
- P-value < .05 implies expecting .05*10000 = 500 false positives
- False discovery rate < .05 implies expecting .05*550 = 27.5 false positives
- Family wise error rate (P (# of false positives ≥ 1)) < .05. See Understanding Family-Wise Error Rate
Statistical significance for genomewide studies by Storey and Tibshirani.
What’s the probability that a significant p-value indicates a true effect?
http://onetipperday.sterding.com/2015/12/my-note-on-multiple-testing.html
A practical guide to methods controlling false discoveries in computational biology by Korthauer, et al 2018, BMC Genome Biology 2019
onlineFDR: an R package to control the false discovery rate for growing data repositories

Suppose \(\displaystyle p_1 \leq p_2 \leq ... \leq p_n\). Then

\(\displaystyle \text{FDR}_i = \text{min}(1, n* p_i/i) \).

So if the number of tests (\(\displaystyle n\)) is large and/or the original p value (\(\displaystyle p_i\)) is large, then FDR can hit the value 1.

However, the simple formula above does not guarantee the monotonicity property from the FDR. So the calculation in R is more complicated. See How Does R Calculate the False Discovery Rate.

Below is the histograms of p-values and FDR (BH adjusted) from a real data (Pomeroy in BRB-ArrayTools).

File:Hist bh.svg

And the next is a scatterplot w/ histograms on the margins from a null data.

File:Scatterhist.svg

q-value

q-value is defined as the minimum FDR that can be attained when calling that feature significant (i.e., expected proportion of false positives incurred when calling that feature significant).

If gene X has a q-value of 0.013 it means that 1.3% of genes that show p-values at least as small as gene X are false positives.

SAM/Significance Analysis of Microarrays

The percentile option is used to define the number of falsely called genes based on 'B' permutations. If we use the 90-th percentile, the number of significant genes will be less than if we use the 50-th percentile/median.

In BRCA dataset, using the 90-th percentile will get 29 genes vs 183 genes if we use median.

Multivariate permutation test

In BRCA dataset, using 80% confidence gives 116 genes vs 237 genes if we use 50% confidence (assuming maximum proportion of false discoveries is 10%). The method is published on EL Korn, JF Troendle, LM McShane and R Simon, Controlling the number of false discoveries: Application to high dimensional genomic data, Journal of Statistical Planning and Inference, vol 124, 379-398 (2004).

String Permutations Algorithm

https://youtu.be/nYFd7VHKyWQ

Empirical Bayes Normal Means Problem with Correlated Noise

Solving the Empirical Bayes Normal Means Problem with Correlated Noise Sun 2018

The package cashr and the source code of the paper

Bayes

Bayes factor

http://www.nicebread.de/what-does-a-bayes-factor-feel-like/

Empirical Bayes method

Naive Bayes classifier

Understanding Naïve Bayes Classifier Using R

MCMC

Speeding up Metropolis-Hastings with Rcpp

offset() function

An offset is a term to be added to a linear predictor, such as in a generalised linear model, with known coefficient 1 rather than an estimated coefficient.
https://www.rdocumentation.org/packages/stats/versions/3.5.0/topics/offset

Offset in Poisson regression

We need to model rates instead of counts
More generally, you use offsets because the units of observation are different in some dimension (different populations, different geographic sizes) and the outcome is proportional to that dimension.

An example from here

Y  <- c(15,  7, 36,  4, 16, 12, 41, 15)
N  <- c(4949, 3534, 12210, 344, 6178, 4883, 11256, 7125)
x1 <- c(-0.1, 0, 0.2, 0, 1, 1.1, 1.1, 1)
x2 <- c(2.2, 1.5, 4.5, 7.2, 4.5, 3.2, 9.1, 5.2)

glm(Y ~ offset(log(N)) + (x1 + x2), family=poisson) # two variables
# Coefficients:
# (Intercept)           x1           x2
#     -6.172       -0.380        0.109
#
# Degrees of Freedom: 7 Total (i.e. Null);  5 Residual
# Null Deviance:	    10.56
# Residual Deviance: 4.559 	AIC: 46.69
glm(Y ~ offset(log(N)) + I(x1+x2), family=poisson)  # one variable
# Coefficients:
# (Intercept)   I(x1 + x2)
#   -6.12652      0.04746
#
# Degrees of Freedom: 7 Total (i.e. Null);  6 Residual
# Null Deviance:	    10.56
# Residual Deviance: 8.001 	AIC: 48.13

Offset in Cox regression

An example from biospear::PCAlasso()

coxph(Surv(time, status) ~ offset(off.All), data = data)
# Call:  coxph(formula = Surv(time, status) ~ offset(off.All), data = data)
#
# Null model
#   log likelihood= -2391.736 
#   n= 500 

# versus without using offset()
coxph(Surv(time, status) ~ off.All, data = data)
# Call:
# coxph(formula = Surv(time, status) ~ off.All, data = data)
#
#          coef exp(coef) se(coef)    z    p
# off.All 0.485     1.624    0.658 0.74 0.46
#
# Likelihood ratio test=0.54  on 1 df, p=0.5
# n= 500, number of events= 438 
coxph(Surv(time, status) ~ off.All, data = data)$loglik
# [1] -2391.702 -2391.430    # initial coef estimate, final coef

Offset in linear regression

Overdispersion

https://en.wikipedia.org/wiki/Overdispersion

Var(Y) = phi * E(Y). If phi > 1, then it is overdispersion relative to Poisson. If phi <1, we have under-dispersion (rare).

Heterogeneity

The Poisson model fit is not good; residual deviance/df >> 1. The lack of fit maybe due to missing data, covariates or overdispersion.

Subjects within each covariate combination still differ greatly.

Consider Quasi-Poisson or negative binomial.

Test of overdispersion or underdispersion in Poisson models

https://stats.stackexchange.com/questions/66586/is-there-a-test-to-determine-whether-glm-overdispersion-is-significant

Negative Binomial

The mean of the Poisson distribution can itself be thought of as a random variable drawn from the gamma distribution thereby introducing an additional free parameter.

Binomial

Generating and modeling over-dispersed binomial data
simstudy package. The final data sets can represent data from randomized control trials, repeated measure (longitudinal) designs, and cluster randomized trials. Missingness can be generated using various mechanisms (MCAR, MAR, NMAR). Analyzing a binary outcome arising out of within-cluster, pair-matched randomization.

Count data

Zero counts

A Method to Handle Zero Counts in the Multinomial Model

Bias

Bias in Small-Sample Inference With Count-Data Models Blackburn 2019

Survival data analysis

See Survival data analysis

Logistic regression

Simulate binary data from the logistic model

https://stats.stackexchange.com/questions/46523/how-to-simulate-artificial-data-for-logistic-regression

set.seed(666)
x1 = rnorm(1000)           # some continuous variables 
x2 = rnorm(1000)
z = 1 + 2*x1 + 3*x2        # linear combination with a bias
pr = 1/(1+exp(-z))         # pass through an inv-logit function
y = rbinom(1000,1,pr)      # bernoulli response variable
 
#now feed it to glm:
df = data.frame(y=y,x1=x1,x2=x2)
glm( y~x1+x2,data=df,family="binomial")

Building a Logistic Regression model from scratch

https://www.analyticsvidhya.com/blog/2015/10/basics-logistic-regression

Odds ratio

Calculate the odds ratio from the coefficient estimates; see this post.

require(MASS)
N  <- 100               # generate some data
X1 <- rnorm(N, 175, 7)
X2 <- rnorm(N,  30, 8)
X3 <- abs(rnorm(N, 60, 30))
Y  <- 0.5*X1 - 0.3*X2 - 0.4*X3 + 10 + rnorm(N, 0, 12)

# dichotomize Y and do logistic regression
Yfac   <- cut(Y, breaks=c(-Inf, median(Y), Inf), labels=c("lo", "hi"))
glmFit <- glm(Yfac ~ X1 + X2 + X3, family=binomial(link="logit"))

exp(cbind(coef(glmFit), confint(glmFit)))<br />

AUC

A small introduction to the ROCR package

       predict.glm()             ROCR::prediction()     ROCR::performance()
glmobj ------------> predictTest -----------------> ROCPPred ---------> AUC
newdata                labels

Medical applications

Subgroup analysis

Other related keywords: recursive partitioning, randomized clinical trials (RCT)

Thinking about different ways to analyze sub-groups in an RCT
Tutorial in biostatistics: data-driven subgroup identification and analysis in clinical trials I Lipkovich, A Dmitrienko - Statistics in medicine, 2017
Personalized medicine:Four perspectives of tailored medicine SJ Ruberg, L Shen - Statistics in Biopharmaceutical Research, 2015
Berger, J. O., Wang, X., and Shen, L. (2014), “A Bayesian Approach to Subgroup Identification,” Journal of Biopharmaceutical Statistics, 24, 110–129.
Change over time is not "treatment response"

Interaction analysis

Goal: assessing the predictiveness of biomarkers by testing their interaction (strength) with the treatment.
Evaluation of biomarkers for treatment selection usingindividual participant data from multiple clinical trials Kang et al 2018
http://www.stat.purdue.edu/~ghobbs/STAT_512/Lecture_Notes/ANOVA/Topic_27.pdf#page=15. For survival data, y-axis is the survival time and B1=treatment, B2=control and X-axis is treatment-effect modifying score. But as seen on page16, the effects may not be separated.
Identification of biomarker-by-treatment interactions in randomized clinical trials with survival outcomes and high-dimensional spaces N Ternès, F Rotolo, G Heinze, S Michiels - Biometrical Journal, 2017
Designing a study to evaluate the benefitof a biomarker for selectingpatient treatment Janes 2015
A visualization method measuring theperformance of biomarkers for guidingtreatment decisions Yang et al 2015. Predictiveness curves were used a lot.
Combining Biomarkers to Optimize Patient TreatmentRecommendations Kang et al 2014. Several simulations are conducted.
An approach to evaluating and comparing biomarkers for patient treatment selection Janes et al 2014
A Framework for Evaluating Markers Used to Select Patient Treatment Janes et al 2014
Tian, L., Alizaden, A. A., Gentles, A. J., and Tibshirani, R. (2014) “A Simple Method for Detecting Interactions Between a Treatment and a Large Number of Covariates,” and the book chapter.
Statistical Methods for Evaluating and Comparing Biomarkers for Patient Treatment Selection Janes et al 2013
Assessing Treatment-Selection Markers using a Potential Outcomes Framework Huang et al 2012
Methods for Evaluating Prediction Performance of Biomarkers and Tests Pepe et al 2012

Measuring the performance of markers for guiding treatment decisions by Janes, et al 2011.

cf <- c(2, 1, .5, 0)
f1 <- function(x) { z <- cf[1] + cf[3] + (cf[2]+cf[4])*x; 1/ (1 + exp(-z)) }
f0 <- function(x) { z <- cf[1] + cf[2]*x; 1/ (1 + exp(-z)) }
par(mfrow=c(1,3))
curve(f1, -3, 3, col = 'red', ylim = c(0, 1), 
      ylab = '5-year DFS Rate', xlab = 'Marker A/D Value', 
      main = 'Predictiveness Curve', lwd = 2)
curve(f0, -3, 3, col = 'black', ylim = c(0, 1),
      xlab = '', ylab = '', lwd = 2, add = TRUE)
legend(.5, .4, c("control", "treatment"), 
       col = c("black", "red"), lwd = 2)

cf <- c(.1, 1, -.1, .5)
curve(f1, -3, 3, col = 'red', ylim = c(0, 1), 
      ylab = '5-year DFS Rate', xlab = 'Marker G Value', 
      main = 'Predictiveness Curve', lwd = 2)
curve(f0, -3, 3, col = 'black', ylim = c(0, 1),
      xlab = '', ylab = '', lwd = 2, add = TRUE)
legend(.5, .4, c("control", "treatment"), 
       col = c("black", "red"), lwd = 2)
abline(v= - cf[3]/cf[4], lty = 2)

cf <- c(1, -1, 1, 2)
curve(f1, -3, 3, col = 'red', ylim = c(0, 1), 
      ylab = '5-year DFS Rate', xlab = 'Marker B Value', 
      main = 'Predictiveness Curve', lwd = 2)
curve(f0, -3, 3, col = 'black', ylim = c(0, 1),
      xlab = '', ylab = '', lwd = 2, add = TRUE)
legend(.5, .85, c("control", "treatment"), 
       col = c("black", "red"), lwd = 2)
abline(v= - cf[3]/cf[4], lty = 2)

File:PredcurveLogit.svg

An Approach to Evaluating and Comparing Biomarkers for Patient Treatment Selection The International Journal of Biostatistics by Janes, 2014. Y-axis is risk given marker, not P(T > t0|X). Good details.
Gunter, L., Zhu, J., and Murphy, S. (2011), “Variable Selection for Qualitative Interactions in Personalized Medicine While Controlling the Family-Wise Error Rate,” Journal of Biopharmaceutical Statistics, 21, 1063–1078.

Statistical Learning

Elements of Statistical Learning Book homepage
From Linear Models to Machine Learning by Norman Matloff
10 Free Must-Read Books for Machine Learning and Data Science
10 Statistical Techniques Data Scientists Need to Master
1. Linear regression
2. Classification: Logistic Regression, Linear Discriminant Analysis, Quadratic Discriminant Analysis
3. Resampling methods: Bootstrapping and Cross-Validation
4. Subset selection: Best-Subset Selection, Forward Stepwise Selection, Backward Stepwise Selection, Hybrid Methods
5. Shrinkage/regularization: Ridge regression, Lasso
6. Dimension reduction: Principal Components Regression, Partial least squares
7. Nonlinear models: Piecewise function, Spline, generalized additive model
8. Tree-based methods: Bagging, Boosting, Random Forest
9. Support vector machine
10. Unsupervised learning: PCA, k-means, Hierarchical
15 Types of Regression you should know

LDA (Fisher's linear discriminant), QDA