# Examining Website Pathing Data Using Markov Chains

A markov model can be used to examine a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Let’s define a stochastic process $(X_{n}, n = 0, 1, 2, ...)$ that takes on a finite number of possible values which are nonnegative integers. Each state, $X_{n}$, represents it’s value in time period $n$. If the probability of being in $X_{n+1}$ is dependent on $X_{n}$, it’s refered to as the first-order Markov property. We are interested in estimating $P_{ij}$, which is the fixed probability that $X$ at time $i$ will be followed by state $j$. These $n$ step transition probabilities are calculated through the Chapman-Kolmogorov equations, which relates the joint probability distributions of different sets of coordinates on a stochastic process. Markov chains are generally represented as a state diagram or transition matrix where every row of the matrix, $P$, is a conditional probability mass function.

Let’s consider an example using website pathing data from an ecommerce website. The set of possible outcomes, or sample space, is defined below. For this example, $S_{i}$ takes the values of each page on the site. This likely violates the Markov property, given that pages on an ecommerce website aren’t generally dependent on the previous page visited, but let’s proceed anyways.

$S = (Home, About, Shoes, Denim, Cart)$

Given a series of clickstreams, a markov chain can be fit in order to predict the next page visited. Below are the state diagram and transition matrix for this data. It suggests that from the home state, there is a 83% probability that a visit to the shoes state will be next.

To be honest, I’m not certain whether is the best technique to model how consumers utilize a website. Markov chains seem to be promising, but I’m a little uneasy about whether our assumptions are met in such scenarios. If you have ideas for such problems, please comment below.

# Statistics Refresher

Let’s face it, a good statistics refresher is always worthwhile. There are times we all forget basic concepts and calculations. Therefore, I put together a document that could act as a statistics refresher and thought that I’d share it with the world. This is part one of a two part document that is still being completed. This refresher is based on Principles of Statistics by Balmer and Statistics in Plain English by Brightman.

### The Two Concepts of Probability

#### Statistical Probability

• Statistical probability pertains to the relative frequency with which an event occurs in the long run.
• Example:
Let’s say we flip a coin twice. What is the probability of getting two heads?
If we flip a coin twice, there are four possible outcomes, $[(H,H), (H,T), (T,H), (T,T)]$.
Therefore, the probability of flipping two heads is $\frac{(H,H)}{N} = \frac{1}{2}*\frac{1}{2} = \frac{1}{4}$

#### Inductive Probability

• Inductive probability pertains to the degree of belief which is reasonable to place on a proposition given evidence.
• Example:
I’m $95\%$ certain that the answer to $1 + 1$ is between $1.5$ and $2.5$.

### The Two Laws of Probability

• If $A$ and $B$ are mutually exclusive events, the probability that either $A$ or $B$ will occur is equal to the sum of their separate probabilities.

$\displaystyle P(A \space or \space B) = P(A) + P(B)$

#### Law of Multiplication

• If $A$ and $B$ are two events, the probability that both $A$ and $B$ will occur is equal to the probability that $A$ will occur multiplied by the conditional probability that $B$ will occur given that $A$ has occured.

$P(A \space and \space B) = P(A) * P(B|A)$

#### Conditional Probability

• The probability of $B$ given $A$, or $P(B|A)$, is the probability that $B$ will occur if we consider only those occasionson which $A$ also occurs. This is defined as $\frac{n(A \space and \space B)}{n(A)}$.

### Random Variables and Probability Distributions

#### Discrete Variables

• Variables which arise from counting and can only take integral values $(0, 1, 2, \ldots)$.
• A frequency distribution represents the amount of occurences for all the possible values of a variable. This can be represented in a table or graphically as a probability distribution.
• Associated with any discrete random variable, $X$, is a corresponding probability function which tells us the probability with which $X$ takes any value. The particular value that $X$ can take is characterized by $x$. Based on $x$, the probability that $X$ will take can be calculated. This measure is the probability function and is defined by $P(x)$.
• The cumulative probability function specifies the probability that $X$ is less than or equal to some particular value, $x$. This is denoted by $F(x)$. The cumulative probability function can be calculated by summing the probabilities of all values less than or equal to $x$.

$F(x) = Prob[X \leq x]$

$F(x) = P(0) + P(1) + \ldots + P(x) = \sum_{u \leq x} p(u)$

#### Continuous Variables

• Variables which arise from measuring and can take any value within a given range.
• Continuous variables are best graphically represented by a histogram, where the area of each rectangle represents the proportion of observations falling in that interval.
• The probability density function, $f(x)$, refers to the smooth continuous curve that is used to describe the relative likelihood a random variable to take on a given value. $f(x)$ can also be used to show the probability that the random variable will lie between $x_1$ and $x_2$.
• A continuous probability distribution can also be represented by its cumulative probability function, $f(x)$. which specified the probability that $X$ is less than or equal to $x$.
• A continuous random variable is said to be uniformly distributed between $0$ and $1$ if it is equally likely to lie anywhere in this interval but cannot lie outside it.

#### Multivariate Distributions

• The joint frequency distribution of two random variables is called a bivariate distribution. $P(x,y)$ denotes the probability that simultaneously $X$ will be $x$ and $Y$ will be $y$. This is expressed through a bivariate distribution table.

$P(x,y) = Prob[X == x \space and \space Y == y]$

• In a bivariate distribution table, the right hand margin sums the probabilities in different rows. It expresses the overall probability distribution of $x$, regardless of the value of $y$.

$p(x) = Prob[X == x] = \sum_{y} p(x,y)$

• In a bivariate distribution table, the bottom margin sums the probabilities in different columns. It expresses the overall probability distribution of $y$, regardless of the value of $x$.

$p(y) = Prob[Y == y] = \sum_{x} p(x,y)$

### Properties of Distributions

#### Measures of Central Tendancy

• The mean is measured by taking the sum divided by the number of observations.

$\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} = \sum_{i=1}^n \frac{x_i}{n}$

• The median is the middle observation in a series of numbers. If the number of observations are even, then the two middle observations would be divided by two.
• The mode refers to the most frequent observation.
• The main question of interest is whether the sample mean, median, or mode provides the most accurate estimate of central tendancy within the population.

#### Measures of Dispersion

• The standard deviation of a set of observations is the square root of the average of the squared deviations from the mean. The squared deviations from the mean is called the variance.

#### The Shape of Distributions

• Unimodal distributions have only one peak while multimodal distributions have several peaks.
• An observation that is skewed to the right contains a few large values which results in a long tail towards the right hand side of the chart.
• An observation that is skewed to the left contains a few small values which results in a long tail towards the left hand side of the chart.
• The kurtosis of a distribution refers to the degree of peakedness of a distribution.

### The Binomial, Poisson, and Exponential Distributions

#### Binomial Distribution

• Think of a repeated process with two possible outcome, failure ($F$) and success ($S$). After repeating the experiment $n$ times, we will have a sequence of outcomes that include both failures and successes, $SFFFSF$. The primary metric of interest is the total number of successes.
• What is the probability of obtaining $x$ successes and $n-x$ failures in $n$ repetitions of the experiment?

#### Poisson Distribution

• The poisson distribution is the limiting form of the binomial distribution when there are a large number of trials but only a small probability of success at each of them.

#### Exponential Distribution

• A continuous, positive random variable is said to follow an exponential distribution if its probability density function decreases as the values of $x$ go from $0$ to $\infty$. The probability declines from its highest levels at the initial values of $x$.

### The Normal Distribution

#### Properties of the Normal Distribution

• The real reason for the importance of the normal distribution lies in the central limit theorem, which states that the sum of a large number of independent random variables will be approximately normally distributed regardless of their individual distributions.
• A normal distribution is defined by its mean, $\mu$, and standard deviation, $\sigma$. A change in the mean shifts the distribution along the x-axis. A change in the standard deviation flattens it or compresses it while leaving its centre in the same position. The totral area under the curve is one and the mean is at the middle and divides the area into halves.
• One standard deviation above and below the mean of a normal distribution will include 68% of the observations for that variable. For two standard deviates, that value will be 95%, and for three standard deviations, that value will be 99%.

There you have it, a quick review of basic concepts in statistics and probability. Please leave comments or suggestions below. If you’re looking to hire a marketing scientist, please contact me at mathewanalytics@gmail.com

# Introduction to the RMS Package

The rms package offers a variety of tools to build and evaluate regression models in R. Originally named ‘Design’, the package accompanies the book “Regression Modeling Strategies” by Frank Harrell, which is essential reading for anyone who works in the ‘data science’ space. Over the past year or so, I have transitioned my personal modeling scripts to rms as it makes things such as bootstrapping, model validation, and plotting predicted probabilities easier to do. While the package is fairly well documented, I wanted to put together a ‘simpler’ and more accessible introduction that would explain to R-beginners how they could start using the rms package. For those with limited statistics training, I strongly suggest reading “Clinical Prediction Models” and working your way up to “Regression Modeling Strategies”.

We start this introduction to the rms package with the datadist function, which computes statistical summaries of predictors to automate estimation and plotting of effects. The user will generally supply the final data frame to the datadist function and set the data distribution using the options function. Note that if you modify the data in your data frame, then you will need to reset the distribution summaries using datadist.


my.df = data.frame(one=c(rnorm(100)), two=c(rnorm(100)), y=rnorm(100))

my.df_new = subset(my.df, two >= 0.05)
ddd



The main functions to estimate models in rms are ols for linear models and lrm for logistic regression or ordinal logistic regression. There are also a few other functions for performing survival analysis,but they will not be covered in this post.


getHdata(prostate)



Using the prostate data, we built a linear model using ordinary least squares estimation. The argument x and y must be set to true if we plan on evaluate the model in later stages using the validate and calibrate functions. Because we haven’t altered the default contrasts or incorporated any smoothing splines, the coefficients and standard errors should be identical to the results of lm. Use the model variable name to see the estimates from the model and use the summary function to get an overview of the effects of each predictor on the response variable. One important thing to note that the effect point estimates in the summary.rms output relate to the estimated effect of an inter-quartile range increase in the predictor variable.


lmod = ols(wt ~ age + sbp + rx, data=prostate, x=TRUE, y=TRUE)
lmod
summary(lmod)
summary(lmod, age=c(50,70))



This may not seem like anything to write home about. But what makes the rms package special is that it makes the modeling process significantly easier. For the above linear regression model, let’s plot the predicted values and perform internal bootstrapped validation of the model. In the following code, the validate function is used to assess model fit and calibrate is used to assess if the observed responses agree with predicted responses.


plot(anova(lmod), what='proportion chisq') # relative importance
plot(Predict(lmod)) # predicted values
rms::validate(lmod, method="boot", B=500) # bootstrapped validation
my.calib <- rms::calibrate(lmod, method="boot", B=500) # model calibration
plot(my.calib, las=1)
vif(lmod) # test for multicolinearity
Predict(lmod)



Let us now build a logistic regression model using the lrm function, plot the expected probabilities, and evaluate the model. We also use the pentrace function to perform logistic regression with penalized maximum likelihood estimation.


mod1 = lrm(as.factor(bm) ~ age + sbp + rx, data=prostate, x=TRUE, y=TRUE)
mod1
summary(mod1)

plot(anova(mod1), what='proportion chisq') # relative importance
plot(Predict(mod1, fun=plogis)) # predicted values
rms::validate(mod1, method="boot", B=500) # bootstrapped validation
my.calib <- rms::calibrate(mod1, method="boot", B=500) # model calibration
plot(my.calib, las=1)

penalty <- pentrace(mod1, penalty=c(0.5,1,2,3,4,6,8,12,16,24), maxit=25)
mod1_pen <- update(mod1, penalty=penalty$penalty) effective.df(mod1_pen) mod1_pen  There you have it, a very basic introduction to the rms package for beginners to the R programming language. Once again, I strongly suggest that readers who are not trained statisticians should read and fully comprehend “Clinical Prediction Models” and “Regression Modeling Strategies” by Frank Harrell. You can also access a number of handouts and lecture notes at here. # Extract Google Trends Data with Python Anyone who has regularly worked with Google Trends data has had to deal with the slightly tedious task of grabbing keyword level data and reformatting the spreadsheet provided by Google. After looking for a seamless way to pull the data, I came upon the PyTrends library on GitHub, and sought to put together some quick user defined functions to manage the task of pulling daily and weekly trends data.  # set working directory (location where code is) import os os.chdir("path") import re import csv import time import pandas as pd from random import randint from GT_Automation_Code import pyGTrends # set gmail credentials and path to extract data google_username = "*****@gmail.com" google_password = "*****" Daily_Data = [ ] # define daily pull code def GT_Daily_Run(keys): path = 'path' # connect to Google connector = pyGTrends(google_username, google_password) # make request connector.request_report(keys, date="today 90-d", geo="US") # wait a random amount of time between requests to avoid bot detection time.sleep(randint(5, 10)) # download file connector.save_csv(path, '_' + "GT_Daily" + '_' + keys.replace(' ', '_')) name = path + '_' + "GT_Daily" + '_' + keys.replace(' ', '_') with open(name + '.csv', 'rt') as csvfile: csvReader = csv.reader(csvfile) data = [] for row in csvReader: if any('2015' in s for s in row): data.append(row) day_df = pd.DataFrame(data) cols = ["Day", keys] day_df.columns = [cols] Daily_Data.append(day_df) keywords = ['soccer', 'football', 'baseball'] map(lambda x: GT_Daily_Run(x), keywords) rge = [Daily_Data[0], Daily_Data[1], Daily_Data[2]] df_final_daily = reduce(lambda left,right: pd.merge(left,right, on='Day'), rge) df_final_daily = df_final_daily.loc[:, (df_final_daily != "0").any(axis=0)] df_final_daily.to_csv("Daily_Trends_Data.csv", index=False) Weekly_Data = [ ] # define weekly pull code def GT_Weekly_Run(keys): path = 'path' # connect to Google connector = pyGTrends(google_username, google_password) # make request connector.request_report(keys, geo="US") # wait a random amount of time between requests to avoid bot detection time.sleep(randint(5, 10)) # download file connector.save_csv(path, '_' + "GT_Weekly" + '_' + keys.replace(' ', '_')) name = path + '_' + "GT_Weekly" + '_' + keys.replace(' ', '_') with open(name + '.csv', 'rt') as csvfile: csvReader = csv.reader(csvfile) data = [] datex = re.compile('(19|20)dd-(0[1-9]|1[012])-(0[1-9]|[12][0-9]|3[01])') for row in csvReader: if datex.search(str(row)): data.append(row) week_df = pd.DataFrame(data) cols = ["Week", keys] week_df.columns = [cols] Weekly_Data.append(week_df) map(lambda x: GT_Weekly_Run(x), keywords) rge = [Weekly_Data[0], Weekly_Data[1], Weekly_Data[2]] df_final_weekly = reduce(lambda left,right: pd.merge(left,right, on='Week'), rge) df_final_weekly = df_final_weekly.loc[:, (df_final_weekly != "0").any(axis=0)] df_final_weekly.to_csv("Weekly_Trends_Data.csv", index=False)  # Batch Forecasting in R Given a data frame with multiple columns which contain time series data, let’s say that we are interested in executing an automatic forecasting algorithm on a number of columns. Furthermore, we want to train the model on a particular number of observations and assess how well they forecast future values. Based upon those testing procedures, we will estimate the full model. This is a fairly simple undertaking, but let’s walk through this task. My preference for such procedures is to loop through each column and append the results into a nested list. First, let’s create some data. ddat <- data.frame(date = c(seq(as.Date("2010/01/01"), as.Date("2010/03/02"), by=1)), value1 = abs(round(rnorm(61), 2)), value2 = abs(round(rnorm(61), 2)), value3 = abs(round(rnorm(61), 2))) head(ddat) tail(ddat) We want to forecast future values of the three columns. Because we want to save the results of these models into a list, lets begin by creating a list that contains the same number of elements as our data frame. lst.names <- c(colnames(data)) lst <- vector("list", length(lst.names)) names(lst) <- lst.names lst I’ve gone ahead and written a user defined function that handles the batch forecasting process. It takes two arguments, a data frame and default argument which specifies the number of observations that will be used in the training set. The model estimates, forecasts, and diagnostic measures will be saved as a nested list and categorized under the appropriate variable name. batch <- function(data, n_train=55){ lst.names <- c(colnames(data)) lst <- vector("list", length(lst.names)) names(lst) <- lst.names for( i in 2:ncol(data) ){ lst[[1]][["train_dates"]] <- data[1:(n_train),1] lst[[1]][["test_dates"]] <- data[(n_train+1):nrow(data),1] est <- auto.arima(data[1:n_train,i]) fcas <- forecast(est, h=6)$mean
acc <- accuracy(fcas, data[(n_train+1):nrow(data),i])
fcas_upd <- data.frame(date=data[(n_train+1):nrow(data),1], forecast=fcas,                           actual=data[(n_train+1):nrow(data),i])

lst[[i]][["estimates"]] <- est
lst[[i]][["forecast"]] <- fcas
lst[[i]][["forecast_f"]] <- fcas_upd
lst[[i]][["accuracy"]] <- acc

cond1 = diff(range(fcas[1], fcas[length(fcas)])) == 0
cond2 = acc[,3] >= 0.025

if(cond1|cond2){

mfcas = forecast(ma(data[,i], order=3), h=5)
lst[[i]][["moving_average"]] <- mfcas

} else {

est2 <- auto.arima(data[,i])
fcas2 <- forecast(est, h=5)\$mean

lst[[i]][["estimates_full"]] <- est2
lst[[i]][["forecast_full"]] <- fcas2

}
}
return(lst)
}

batch(ddat)

This isn’t the prettiest code, but it gets the job done. Note that lst was populated within a function and won’t be available in the global environment. Instead, I chose to simply print out the contents of the list after the function is evaluated.

# R Programming Notes

I’ve been on a note taking binge recently. This post covers a variety of topics related to programming in R. The contents were gathered from many sources and structured in such a way that it provided the author with a useful reference guide for understanding a number of useful R functions.

DO.CALL

The do.call function executes a function call on a list of arguments.

do.call("R_Function", "List_of_Arguments")

This is equivilant to telling R which arguments the function should operate on.

R_Function( "List_of_Arguments" ){
...
}

Consider the following list with four elements. We can use this function to find the total sum across all list elements or bind the rows into a data frame.

x1 <- c(1,2,5)
x2 <- c(1,3,6)
x3 <- c(1,4,7)
x4 <- c(1,5,8)

do.call(sum, list(x1,x2,x3,x4))  # sum all list elements
do.call(rbind, list(x1,x2,x3,x4))  # rbind the list elements

Let’s consider a scenario where we have a small data frame and want to run a general linear model on different combintions of attributes. One solution to this problem is to create the formula object as a list within a function, and then utilize do.call to run a linear model on the desired attribute.

dat <- data.frame(x1 = rnorm(100, m=50), x2 = rnorm(100, m=50), x3 = rnorm(100, m=50), y = rnorm(100))

new_mod <- function(form){
lstt = list(formula=as.formula(paste("y ~ ", form, sep="")), data=as.name("dat"))
summary(do.call("lm", lstt))
}

new_mod("x1")
new_mod("x2")
new_mod("x3")

EVAL

The eval function evaluates an expression, an object that represents an action that can be performed in R. An expression is different from an operation, which refers to the execution of an operation. In the following example, we assigned a value to a variable and performed an operation using that variable.

x <- 4
y <- x * 10
y

The expression and quote functions are used to takes an expression as an argument and returns an expression without evaluation.

ee = expression(y~x1)
ee

z <- quote(y <- x * 10)
z

is.expression(ee)
is.call(z)

In the above example, z is referred to as a call object. They are usually created with the call function. For example, the following code pieces together commands into a call object and evaluates them.

mycall <- call("lm", quote(y ~ x1), data=quote(dat))
mod <- eval(mycall)
summary(mod)

SUBSTITUTE

Another common procedure is to replace certain variables within a user defined function. This can be achieved with the substitute function. The below code replaces x1 and x2 in the expression with the values from the list.

replace <- list(x1 = as.name("alphabet"), x2 = as.name("zoology"))
substitute(expression(x1 + x2 + log(x1) + x3), replace)

SETNAMES

There are a number of commands for working within column and row names in R. It’s generally suggested that the setNames function be used when modifying data structure names within a function.

names_new <- c("one","two","three","four")
new_dat <- expression(data.frame(x1 = rnorm(100, m=50), x2 = rnorm(100, m=50), x3 = rnorm(100, m=50), y = rnorm(100)))

my_lst <- list(lname = as.name("xls"))
setNames(my_lst, "x1")

When working with data within a function, it will often be useful to write code that creates the names and data structure, and then evaluates the pieces together.

MISSING

Use the missing function to check whether an argument was supplied by the user.

func <- function(x){
if(missing(x)){
x = 5
}
y = x*x; y
}

func(2)
func()

STOP

The stop function is used to halt a function, and is usually used when a particular condition is met.

guess <- function(n){
if( n >= 6 ){
stop("WRONG!")
}}

guess(5)
guess(10)

WARNING

Use the warning function to throw a comment when a condition is met. It does not halt the function.

guess <- function(n){
if( n >= 6 ){
warning("BE CAREFUL!")
}}

guess(5)
guess(10)

ELIPSES (…)

When functions are evaluated, R scans each function argument it can understand. When ellipsis are added as a function argument, it allows for other arguments to be passed into the function. This technique is frequently used when plotting and should be used when the function is designed to take any number of named or unnamed arguments.

plotter <- function(x, y, ...){
plot(x, y, ...)
}

To make use of ellipsis, it’s suggested that the user scan through the … and turn them into a list. This is because some of the arguments in dots may be intended for several different functions.

For those of us who received statistical training outside of statistics departments, it often emphasized procedures over principles. This entailed that we learned about various statistical techniques and how to perform analysis in a particular statistical software, but glossed over the mechanisms and mathematical statistics underlying these practices. While that training methodology (hereby referred to as the ‘heuristic method’) has value, it has many drawbacks when the ultimate goal is to perform sound statistical analysis that is valid and thorough. Even in my current role as a data scientist at a technology company in the San Francisco Bay Area, I have had to go back and understand various procedures and metrics instead of just “doing data analysis”.

Given this realization, I have dedicated hours of time outside of work over the last couple years to “re-training” myself on many of the important concepts in both descriptive and inferential statistics. This post will give brief mention to the books that have been most useful is helping me develop a fuller understanding of the statistical sciences. These books have also helped me fill in the gaps and deficiencies from my statistical training in university and graduate school. Furthermore, these are the texts that I often revisit when I need a reference on some statistical topic of interest. This is at a minimum a a six year journey, so I have a long way to go until I am able to stand solidly in my understanding of statistics. While I am sacrificing a lot of my free time to this undertaking, it will certainly improve my knowledge and help prepare me for graduate school (PhD) in biostatistics, which I hope to attend in around five to six years.[1]

Please note that I am not taking issue with the ‘heuristic method’ of statistical training. It certainly has its place and provides students with the immediate knowledge required to satisfactorily prepare for work in private industry. In fact, I prefer the ‘heuristic method’ and still rely on straight forward rules in my day to day work as that ensures that best practices are followed and satisfactory analysis is performed. Furthermore, I certainly believe that it is superior to the hack-ey nature of data mining and data science education, but that is a different story.

#### Fundamentals:

Statistics in Plain English – Urdan
Clear, concise, and covers all the fundamental items that one would need to know. Everything from descriptive statistics to linear regression are covered, with many good examples. Even if you never use ANOVA or factor analysis, this is a good book to review and one that I strongly recommend to people who are interested in data science.

Principles of Statistics – Balmer
This is a classic text that offers a good treatment of probability theory, distributions, and statistical inference. The text contains a bit more math than ‘Statistics in Plain English’, so I think it should be read after completing the previous book.

Fundamentals of Modern Statistical Methods – Wilcox
This book reviews ‘traditional’ parametric statistics and provides a good overview of robust statistical methods. There is a fair amount on the historical evolution of various techniques, and I found that a bit unnecessary. But overall, this is still a solid introductory text to learn about statistical inference using robust techniques.

#### Regression Analysis:

Mostly Harmless Econometrics – Angrist
While I don’t regularly work with instrumental variables, generalized methods of moments, or regression discontinuity, this book is a great high level introduction to econometrics. The chapters on regression theory and quantile regression are phenomenal.

Regression Modeling Strategies – Harrell
This is my most referenced book and the one that really helped in my overall development as an applied statistician. All the important topics are covered, from imputation, regression splines, and so forth. This book includes R code for performing analysis using the RMS package. I end up citing this book quite a lot. For example, in a recent work email, I mentioned that Harrell “also says on page 61 that “narrowly distributed predictor variables will require higher sample sizes.”” Essential reading in my opinion.

Data Analysis Using Regression and Multilevel/Hierarchical Models – Gelman and Hill
The first half of this book cover statistical inference using single level models and the second half is dedicated to multilevel methods. Given that I am rarely work with panel data, I use the first half of this book a reference for things that I may need a quick refresher on. It is very accessible and has plenty of examples with R code.

Semiparametric Regression for the Social Sciences – Keele
This is one of my favorite statistical books. Well written and easy to comprehend, but still rigorous. Covers local regression, splines, and generalized additive models. There is also a solid chapter on the use of bootstrapping with semiparametric and nonparametric models.

#### Statistical Learning:

Statistical Learning from a Regression Perspective – Berk
As a skeptic who is wary of every hype machine, I really enjoyed Berks preface in which he discusses the “dizzying array of new statistical procedures” that have been introduced over the past several decades with “the hype of a big-budget movie.” I got this text for its treatment of topics such as boosting, bagging, random forest, and support vector machines. I will probably need to reread this book several more times before I fully comprehend everything.

#### Time Series:

Time Series: a Biostatistical Introduction – Diggle
The lack of quality time series books is really infuriating. Don’t get me wrong, there are some good texts on forecasting, such as the free online book from Hyndaman. However, I’ve yet to find a really good intermediate level treatment of time series analysis besides this one. Contains good coverage of repeated measurements, ARIMA modeling, and forecasting.

#### Bayesian Methods

Statistical Rethinking – McElreath
While I was introduced to robust techniques and nonparametric statistics in graduate school, there was nothing on Bayesian methods. Due to a fear of the topic, I avoided learning about it until this past year. This book by McElreath has been great as it is very accessible and provides code for understanding various principles. Over the next year, I am hoping to dive deeper into Bayesian techniques and this was a good first step.