Combinations in probability theory and other areas of mathematics refer to a sequence of outcomes where the order does not matter. For example, when you’re ordering a pizza, it doesn’t matter whether you order it with ham, mushrooms, and olives or olives, mushrooms, and ham. You’re getting the same pizza! [Read more…] about Using Combinations to Calculate Probabilities
The law of large numbers states that as the number of trials increases, sample values tend to converge on the expected result. The two forms of this law lay the foundation for both statistics and probability theory.
In this post, I explain both forms of the law, simulate them in action, and explain why they’re crucial for statistics and probability! [Read more…] about Law of Large Numbers
Permutations in probability theory and other branches of mathematics refer to sequences of outcomes where the order matters. For example, 9-6-8-4 is a permutation of a four-digit PIN because the order of numbers is crucial. When calculating probabilities, it’s frequently necessary to calculate the number of possible permutations to determine an event’s probability.
In this post, I explain permutations and show how to calculate the number of permutations both with repetition and without repetition. Finally, we’ll work through a step-by-step example problem that uses permutations to calculate a probability. [Read more…] about Using Permutations to Calculate Probabilities
Spearman’s correlation in statistics is a nonparametric alternative to Pearson’s correlation. Use Spearman’s correlation for data that follow curvilinear, monotonic relationships and for ordinal data. Statisticians also refer to Spearman’s rank order correlation coefficient as Spearman’s ρ (rho).
In this post, I’ll cover what all that means so you know when and why you should use Spearman’s correlation instead of the more common Pearson’s correlation. [Read more…] about Spearman’s Correlation Explained
Proxy variables are easily measurable variables that analysts include in a model in place of a variable that cannot be measured or is difficult to measure. Proxy variables can be something that is not of any great interest itself, but has a close correlation with the variable of interest. [Read more…] about Proxy Variables: The Good Twin of Confounding Variables
The multiplication rule in probability allows you to calculate the probability of multiple events occurring together using known probabilities of those events individually. There are two forms of this rule, the specific and general multiplication rules.
In this post, learn about when and how to use both the specific and general multiplication rules. Additionally, I’ll use and explain the standard notation for probabilities throughout, helping you learn how to interpret it. We’ll work through several example problems so you can see them in action. There’s even a bonus problem at the end! [Read more…] about Multiplication Rule for Calculating Probabilities
Contingency tables are a great way to classify outcomes and calculate different types of probabilities. These tables contain rows and columns that display bivariate frequencies of categorical data. Analysts also refer to contingency tables as crosstabulation (cross tabs), two-way tables, and frequency tables.
Statisticians use contingency tables for a variety of reasons. I love these tables because they both organize your data and allow you to answer a diverse set of questions. In this post, I focus on using them to calculate different types of probabilities. These probabilities include joint, marginal, and conditional probabilities. [Read more…] about Using Contingency Tables to Calculate Probabilities
Probability theory analyzes the likelihood of events occurring. You can think of probabilities as being the following:
- The long-term proportion of times an event occurs during a random process.
- The propensity for a particular outcome to occur.
Common terms for describing probabilities include likelihood, chances, and odds. [Read more…] about Probability Fundamentals
Variance Inflation Factors (VIFs) measure the correlation among independent variables in least squares regression models. Statisticians refer to this type of correlation as multicollinearity. Excessive multicollinearity can cause problems for regression models.
In this post, I focus on VIFs and how they detect multicollinearity, why they’re better than pairwise correlations, how to calculate VIFs yourself, and interpreting VIFs. If you need a refresher about the types of problems that multicollinearity causes and how to fix them, read my post: Multicollinearity: Problems, Detection, and Solutions. [Read more…] about Variance Inflation Factors (VIFs)
In my post about how to interpret p-values, I emphasize that p-values are not an error rate. The number one misinterpretation of p-values is that they are the probability of the null hypothesis being correct.
The correct interpretation is that p-values indicate the probability of observing your sample data, or more extreme, when you assume the null hypothesis is true. If you don’t solidly grasp that correct interpretation, please take a moment to read that post first.
Hopefully, that’s clear.
Unfortunately, one part of that blog post confuses some readers. In that post, I explain how p-values are not a probability, or error rate, of a hypothesis. I then show how that misinterpretation is dangerous because it overstates the evidence against the null hypothesis. [Read more…] about P-Values, Error Rates, and False Positives
The coefficient of variation (CV) is a relative measure of variability that indicates the size of a standard deviation in relation to its mean. It is a standardized, unitless measure that allows you to compare variability between disparate groups and characteristics. It is also known as the relative standard deviation (RSD).
In this post, you will learn about the coefficient of variation, how to calculate it, know when it is particularly useful, and when to avoid it. [Read more…] about Coefficient of Variation in Statistics
When comparing groups in your data, you can have either independent or dependent samples. The type of samples in your design impacts sample size requirements, statistical power, the proper analysis, and even your study’s costs. Understanding the implications of each type of sample can help you design a better study. [Read more…] about Independent and Dependent Samples in Statistics
Having independent and identically distributed (IID) data is a common assumption for statistical procedures and hypothesis tests. But what does that mouthful of words actually mean? That’s the topic of this post! And, I’ll provide helpful tips for determining whether your data are IID. [Read more…] about Independent and Identically Distributed Data (IID)
Moving averages can smooth time series data, reveal underlying trends, and identify components for use in statistical modeling. Smoothing is the process of removing random variations that appear as coarseness in a plot of raw time series data. It reduces the noise to emphasize the signal that can contain trends and cycles. Analysts also refer to the smoothing process as filtering the data. [Read more…] about Using Moving Averages to Smooth Time Series Data
Note: this is a guest post by Alexander Moreno, a Computer Science PhD student at the Georgia Institute of Technology. He blogs at www.boostedml.com
Survival analysis is an important subfield of statistics and biostatistics. These methods involve modeling the time to a first event such as death. In this post we give a brief tour of survival analysis. We first describe the motivation for survival analysis, and then describe the hazard and survival functions. We follow this with non-parametric estimation via the Kaplan Meier estimator. Then we describe Cox’s proportional hazard model and after that Aalen’s additive model. Finally, we conclude with a brief discussion.
Why Survival Analysis: Right Censoring
Modeling first event times is important in many applications. This could be time to death for severe health conditions or time to failure of a mechanical system. If one always observed the event time and it was guaranteed to occur, one could model the distribution directly. For instance, in the non-parametric setting, one could use the empirical cumulative distribution function to estimate the probability of death by some time. In the parametric setting one could do non-negative regression.
However, in some cases one might not observe the event time: this is generally called right censoring. In clinical trials with death as the event, this occurs when one of the following happens. 1) participants drop out of the study 2) the study reaches a pre-determined end time, and some participants have survived until the end 3) the study ends when a certain number of participants have died. In each case, after the surviving participants have left the study, we don’t know what happens to them. We then have the question:
- How can we model the empirical distribution or do non-negative regression when for some individuals, we only observe a lower bound on their event time?
The above figure illustrates right censoring. For participant 1 we see when they died. Participant 2 dropped out, and we know that they survived until then, but don’t know what happened afterwards. For participant 3, we know that they survived until the pre-determined study end, but again don’t know what happened afterwards.
The Survival Function and the Hazard
Two of the key tools in survival analysis are the survival function and the hazard. The survival function describes the probability of the event not having happened by a time . The hazard describes the instantaneous rate of the first event at any time .
More formally, let be the event time of interest, such as the death time. Then the survival function is . We can also note that this is related to the cumulative distribution function via .
For the hazard, the probability of the first event time being in the small interval , given survival up to is . This is illustrated in the following figure.
Rearranging terms and taking limits we obtain
where is the density function of and the second equality follows from applying Bayes theorem. By rearranging again and solving a differential equation, we can use the hazard to compute the survival function via
The key question then is how to estimate the hazard and/or survival function.
Non-Parametric Estimation with Kaplan Meier
In non-parametric survival analysis, we want to estimate the survival function without covariates, and with censoring. If we didn’t have censoring, we could start with the empirical CDF . This equation is a succinct representation of: how many people have died by time ? The survival function would then be: how many people are still alive? However, we can’t answer this question as posed when some people are censored by time .
While we don’t necessarily know how many people have survived by an arbitrary time , we do know how many people in the study are still at risk. We can use this instead. Partition the study time into , where each is either an event time or a censoring time for a participant. Assume that participants can only lapse at observed event times. Let be the number of people at risk at just before time . Assuming no one dies at exactly the same time (no ties), we can look at each time someone died. We say that the probability of dying at that specific time is , and say that the probability of dying at any other time is . We can then say that the probability of surviving at any event time , given survival at previous candidate event times is . The probability of surviving up to a time is then
We call this  the Kaplan Meier estimator. Under mild assumptions, including that participants have independent and identically distributed event times and that censoring and event times are independent, this gives an estimator that is consistent. The next figure gives an example of the Kaplan Meier estimator for a simple case.
Kaplan Meier R Example
In R we can use the Surv and survfit functions from the survival package to fit a Kaplan Meier model. We can also use ggsurvplot from the survminer package to make plots. Here we will use the ovarian cancer dataset from the survival package. We will stratify based on treatment group assignment.
library(survminer) library(survival) kaplan_meier <- Surv(time = ovarian[['futime']], event = ovarian[['fustat']]) kaplan_meier_treatment<-survfit(kaplan_meier~rx,data=ovarian, type='kaplan-meier',conf.type='log') ggsurvplot(kaplan_meier_treatment,conf.int = 'True')
Semi-Parametric Regression with Cox’s Proportional Hazards Model
Kaplan Meier makes sense when we don’t have covariates, but often we want to model how some covariates affect death risk. For instance, how does one’s weight affect death risk? One way to do this is to assume that covariates have a multiplicative effect on the hazard. This leads us to Cox’s proportional hazard model, which involves the following functional form for the hazard:
The baseline hazard describes how the average person’s risk evolves over time. The relative risk describes how covariates affect the hazard. In particular, a unit increase in leads to an increase of the hazard by a factor of .
Because of the non-parametric nuissance term , it is difficult to maximize the full likelihood for directly. Cox’s insight  was that the assignment probabilities given the death times contain most of the information about , and the remaining terms contain most of the information about . The assignment probabilities give the following partial likelihood
We can then maximize this to get an estimator of . In [3,4] they show that this estimator is consistent and asymptotically normal.
Cox Proportional Hazards R Example
In R, we can use the Surv and coxph functions from the survival package. For the ovarian cancer dataset, we notice from the Kaplan Meier example that treatment is not proportional. Under a proportional hazards assumption, the curves would have the same pattern but diverge. However, instead they move apart and then move back together. Further, treatment does seem to lead to different survival patterns over shorter time horizons. We should not use it as a covariate, but we can stratify based on it. In R we can regress on age and presence of residual disease.
cox_fit <- coxph(Surv(futime, fustat) ~ age + ecog.ps+strata(rx), data=ovarian) summary(cox_fit)
which gives the following results
Call: coxph(formula = Surv(futime, fustat) ~ age + ecog.ps + strata(rx), data = ovarian) n= 26, number of events= 12 coef exp(coef) se(coef) z Pr(>|z|) age 0.13853 1.14858 0.04801 2.885 0.00391 ** ecog.ps -0.09670 0.90783 0.62994 -0.154 0.87800 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 exp(coef) exp(-coef) lower .95 upper .95 age 1.1486 0.8706 1.0454 1.262 ecog.ps 0.9078 1.1015 0.2641 3.120 Concordance= 0.819 (se = 0.058 ) Likelihood ratio test= 12.71 on 2 df, p=0.002 Wald test = 8.43 on 2 df, p=0.01 Score (logrank) test = 12.24 on 2 df, p=0.002
this suggests that age has a significant multiplicative effect on death, and that a one year increase in age increases instantaneous risk by a factor of 1.15.
Aalen’s Additive Model
Cox regression makes two strong assumptions: 1) that covariate effects are constant over time 2) that effects are multiplicative. Aalen’s additive model  relaxes the first, and replaces the second with the assumption that effects are additive. Here the hazard takes the form
As this is a linear model, we can estimate the cumulative regression functions using a least squares type procedure.
Aalen’s Additive Model R Example
In R we can use the timereg package and the aalen function to estimate cumulative regression functions, which we can also plot.
library(timereg) data(sTRACE) # Fits Aalen model out<-aalen(Surv(time,status==9)~age+sex+diabetes+chf+vf, sTRACE,max.time=7,n.sim=100) summary(out) par(mfrow=c(2,3)) plot(out)
This gives us
Additive Aalen Model Test for nonparametric terms Test for non-significant effects Supremum-test of significance p-value H_0: B(t)=0 (Intercept) 7.29 0.00 age 8.63 0.00 sex 2.95 0.01 diabetes 2.31 0.24 chf 5.30 0.00 vf 2.95 0.03 Test for time invariant effects Kolmogorov-Smirnov test (Intercept) 0.57700 age 0.00866 sex 0.11900 diabetes 0.16200 chf 0.12900 vf 0.43500 p-value H_0:constant effect (Intercept) 0.00 age 0.00 sex 0.18 diabetes 0.43 chf 0.06 vf 0.02 Cramer von Mises test (Intercept) 0.875000 age 0.000179 sex 0.017700 diabetes 0.041200 chf 0.053500 vf 0.434000 p-value H_0:constant effect (Intercept) 0.00 age 0.00 sex 0.29 diabetes 0.42 chf 0.02 vf 0.05 Call: aalen(formula = Surv(time, status == 9) ~ age + sex + diabetes + chf + vf, data = sTRACE, max.time = 7, n.sim = 100)
The results first test whether the cumulative regression functions are non-zero, and then whether the effects are constant. The plots of the cumulative regression functions are given below.
In this post we did a brief tour of several methods in survival analysis. We first described why right censoring requires us to develop new tools. We then described the survival function and the hazard. Next we discussed the non-parametric Kaplan Meier estimator and the semi-parametric Cox regression model. We concluded with Aalen’s additive model.
 Kaplan, Edward L., and Paul Meier. “Nonparametric estimation from incomplete observations.” Journal of the American statistical association 53, no. 282 (1958): 457-481.
 Cox, David R. “Regression models and life-tables.” In Breakthroughs in statistics, pp. 527-541. Springer, New York, NY, 1992.
 Tsiatis, Anastasios A. “A large sample study of Cox’s regression model.” The Annals of Statistics 9, no. 1 (1981): 93-108.
 Andersen, Per Kragh, and Richard David Gill. “Cox’s regression model for counting processes: a large sample study.” The annals of statistics (1982): 1100-1120.
 Aalen, Odd. “A model for nonparametric regression analysis of counting processes.” In Mathematical statistics and probability theory, pp. 1-25. Springer, New York, NY, 1980.
Failing to reject the null hypothesis is an odd way to state that the results of your hypothesis test are not statistically significant. Why the peculiar phrasing? “Fail to reject” sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? There’s an excellent reason for the odd wording!
In this post, learn what it means when you fail to reject the null hypothesis and why that’s the correct wording. While accepting the null hypothesis sounds more straightforward, it is not statistically correct! [Read more…] about Failing to Reject the Null Hypothesis
Significance levels in statistics are a crucial component of hypothesis testing. However, unlike other values in your statistical output, the significance level is not something that statistical software calculates. Instead, you choose the significance level. Have you ever wondered why?
In this post, I’ll explain the significance level conceptually, why you choose its value, and how to choose a good value. Statisticians also refer to the significance level as alpha (α). [Read more…] about Understanding Significance Levels in Statistics
Outliers are unusual values in your dataset, and they can distort statistical analyses and violate their assumptions. Unfortunately, all analysts will confront outliers and be forced to make decisions about what to do with them. Given the problems they can cause, you might think that it’s best to remove them from your data. But, that’s not always the case. Removing outliers is legitimate only for specific reasons. [Read more…] about Guidelines for Removing and Handling Outliers in Data
Outliers are data points that are far from other data points. In other words, they’re unusual values in a dataset. Outliers are problematic for many statistical analyses because they can cause tests to either miss significant findings or distort real results.
Unfortunately, there are no strict statistical rules for definitively identifying outliers. Finding outliers depends on subject-area knowledge and an understanding of the data collection process. While there is no solid mathematical definition, there are guidelines and statistical tests you can use to find outlier candidates. [Read more…] about 5 Ways to Find Outliers in Your Data