Statistics By Jim

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Using Post Hoc Tests with ANOVA

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Post hoc tests are an integral part of ANOVA. When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all of the group means are equal. However, ANOVA results do not identify which particular differences between pairs of means are significant. Use post hoc tests to explore differences between multiple group means while controlling the experiment-wise error rate.

In this post, I’ll show you what post hoc analyses are, the critical benefits they provide, and help you choose the correct one for your study. Additionally, I’ll show why failure to control the experiment-wise error rate will cause you to have severe doubts about your results. [Read more…] about Using Post Hoc Tests with ANOVA

One-Tailed and Two-Tailed Hypothesis Tests Explained

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Choosing whether to perform a one-tailed or a two-tailed hypothesis test is one of the methodology decisions you might need to make for your statistical analysis. This choice can have critical implications for the types of effects it can detect, the statistical power of the test, and potential errors.

In this post, you’ll learn about the differences between one-tailed and two-tailed hypothesis tests and their advantages and disadvantages. I include examples of both types of statistical tests. In my next post, I cover the decision between one and two-tailed tests in more detail.
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Introduction to Bootstrapping in Statistics with an Example

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Bootstrapping is a statistical procedure that resamples a single dataset to create many simulated samples. This process allows you to calculate standard errors, construct confidence intervals, and perform hypothesis testing for numerous types of sample statistics. Bootstrap methods are alternative approaches to traditional hypothesis testing and are notable for being easier to understand and valid for more conditions.

In this blog post, I explain bootstrapping basics, compare bootstrapping to conventional statistical methods, and explain when it can be the better method. Additionally, I’ll work through an example using real data to create bootstrapped confidence intervals. [Read more…] about Introduction to Bootstrapping in Statistics with an Example

Estimating a Good Sample Size for Your Study Using Power Analysis

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Determining a good sample size for a study is always an important issue. After all, using the wrong sample size can doom your study from the start. Fortunately, power analysis can find the answer for you. Power analysis combines statistical analysis, subject-area knowledge, and your requirements to help you derive the optimal sample size for your study.

Statistical power in a hypothesis test is the probability that the test will detect an effect that actually exists. As you’ll see in this post, both under-powered and over-powered studies are problematic. Let’s learn how to find a good sample size for your study! [Read more…] about Estimating a Good Sample Size for Your Study Using Power Analysis

Understanding Interaction Effects in Statistics

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Interaction effects occur when the effect of one variable depends on the value of another variable. Interaction effects are common in regression analysis, ANOVA, and designed experiments. In this blog post, I explain interaction effects, how to interpret them in statistical designs, and the problems you will face if you don’t include them in your model. [Read more…] about Understanding Interaction Effects in Statistics

Using Log-Log Plots to Determine Whether Size Matters

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Log-log plots display data in two dimensions where both axes use logarithmic scales. When one variable changes as a constant power of another, a log-log graph shows the relationship as a straight line. In this post, I’ll show you why these graphs are valuable and how to interpret them. [Read more…] about Using Log-Log Plots to Determine Whether Size Matters

When Do You Need to Standardize the Variables in a Regression Model?

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Standardization is the process of putting different variables on the same scale. In regression analysis, there are some scenarios where it is crucial to standardize your independent variables or risk obtaining misleading results.

In this blog post, I show when and why you need to standardize your variables in regression analysis. Don’t worry, this process is simple and helps ensure that you can trust your results. In fact, standardizing your variables can reveal essential findings that you would otherwise miss! [Read more…] about When Do You Need to Standardize the Variables in a Regression Model?

Flu Shots, How Effective Are They?

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With the arrival of Fall in the Northern hemisphere, it’s flu season again.

Do you debate getting a flu shot every year? I do get flu shots every year. I realize that they’re not perfect, but I figure they’re a low-cost way to reduce my chances of a crummy week suffering from the flu.

The media report that flu shots have an effectiveness of approximately 68%. But, what does that mean exactly? What is the absolute reduction in risk? Are there long-term benefits?

In this blog post, I explore the effectiveness of flu shots from a statistical viewpoint. We’ll statistically analyze the data ourselves so we can go beyond the simplified accounts that the media presents. I’ll also model the long-term outcomes you can expect with regular flu vaccinations. By the time you finish this post, you’ll have a crystal clear picture of flu shot effectiveness. Some of the results surprised me! [Read more…] about Flu Shots, How Effective Are They?

Understand Precision in Predictive Analytics to Avoid Costly Mistakes

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Precision in predictive analytics refers to how close the model’s predictions are to the observed values. The more precise the model, the closer the data points are to the predictions. When you have an imprecise model, the observations tend to be further away from the predictions, thereby reducing the usefulness of the predictions. If you have a model that is not sufficiently precise, you risk making costly mistakes! [Read more…] about Understand Precision in Predictive Analytics to Avoid Costly Mistakes

How to Choose Between Linear and Nonlinear Regression

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As you fit regression models, you might need to make a choice between linear and nonlinear regression models. The field of statistics can be weird. Despite their names, both forms of regression can fit curvature in your data. So, how do you choose? In this blog post, I show you how to choose between linear and nonlinear regression models. [Read more…] about How to Choose Between Linear and Nonlinear Regression

Comparing Regression Lines with Hypothesis Tests

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How do you compare regression lines statistically? Imagine you are studying the relationship between height and weight and want to determine whether this relationship differs between basketball players and non-basketball players. You can graph the two regression lines to see if they look different. However, you should perform hypothesis tests to determine whether the visible differences are statistically significant. In this blog post, I show you how to determine whether the differences between coefficients and constants in different regression models are statistically significant. [Read more…] about Comparing Regression Lines with Hypothesis Tests

Identifying the Most Important Independent Variables in Regression Models

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You’ve settled on a regression model that contains independent variables that are statistically significant. By interpreting the statistical results, you can understand how changes in the independent variables are related to shifts in the dependent variable. At this point, it’s natural to wonder, “Which independent variable is the most important?” [Read more…] about Identifying the Most Important Independent Variables in Regression Models

Using Data Mining to Select Regression Models Can Create Serious Problems

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Data mining and regression seem to go together naturally. I’ve described regression as a seductive analysis because it is so tempting and so easy to add more variables in the pursuit of a larger R-squared. In this post, I’ll begin by illustrating the problems that data mining creates. To do this, I’ll show how data mining with regression analysis can take randomly generated data and produce a misleading model that appears to have significant variables and a good R-squared. Then, I’ll explain how data mining creates these deceptive results and how to avoid them. [Read more…] about Using Data Mining to Select Regression Models Can Create Serious Problems

Guide to Stepwise Regression and Best Subsets Regression

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Automatic variable selection procedures are algorithms that pick the variables to include in your regression model. Stepwise regression and Best Subsets regression are two of the more common variable selection methods. In this post, I compare how these methods work and which one provides better results. [Read more…] about Guide to Stepwise Regression and Best Subsets Regression

Goodness-of-Fit Tests for Discrete Distributions

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Discrete probability distributions are based on discrete variables, which have a finite or countable number of values. In this post, I show you how to perform goodness-of-fit tests to determine how well your data fit various discrete probability distributions. [Read more…] about Goodness-of-Fit Tests for Discrete Distributions

Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes

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In my house, we love the Mythbusters TV show on the Discovery Channel. The Mythbusters conduct scientific investigations in their quest to test myths and urban legends. In the process, the show provides some fun examples of when and how you should use statistical hypothesis tests to analyze data. [Read more…] about Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes

Making Predictions with Regression Analysis

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If you were able to make predictions about something important to you, you’d probably love that, right? It’s even better if you know that your predictions are sound. In this post, I show how to use regression analysis to make predictions and determine whether they are both unbiased and precise. [Read more…] about Making Predictions with Regression Analysis

Curve Fitting using Linear and Nonlinear Regression

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In regression analysis, curve fitting is the process of specifying the model that provides the best fit to the specific curves in your dataset. Curved relationships between variables are not as straightforward to fit and interpret as linear relationships. [Read more…] about Curve Fitting using Linear and Nonlinear Regression

How to Interpret P-values and Coefficients in Regression Analysis

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P-values and coefficients in regression analysis work together to tell you which relationships in your model are statistically significant and the nature of those relationships. The coefficients describe the mathematical relationship between each independent variable and the dependent variable. The p-values for the coefficients indicate whether these relationships are statistically significant. [Read more…] about How to Interpret P-values and Coefficients in Regression Analysis

How to Interpret Adjusted R-Squared and Predicted R-Squared in Regression Analysis

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R-squared tends to reward you for including too many independent variables in a regression model, and it doesn’t provide any incentive to stop adding more. Adjusted R-squared and predicted R-squared use different approaches to help you fight that impulse to add too many. The protection that adjusted R-squared and predicted R-squared provide is critical because too many terms in a model can produce results that you can’t trust. These statistics help you include the correct number of independent variables in your regression model. [Read more…] about How to Interpret Adjusted R-Squared and Predicted R-Squared in Regression Analysis