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
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Heteroscedasticity in Regression Analysis
Heteroscedasticity means unequal scatter. In regression analysis, we talk about heteroscedasticity in the context of the residuals or error term. Specifically, heteroscedasticity is a systematic change in the spread of the residuals over the range of measured values. Heteroscedasticity is a problem because ordinary least squares (OLS) regression assumes that all residuals are drawn from a population that has a constant variance (homoscedasticity).
To satisfy the regression assumptions and be able to trust the results, the residuals should have a constant variance. In this blog post, I show you how to identify heteroscedasticity, explain what produces it, the problems it causes, and work through an example to show you several solutions. [Read more…] about Heteroscedasticity in Regression Analysis
Statistics, Old Love Letters, and Changing Times
Have you ever seen your present reflected in an object from the past? This summer I’ve discovered glimpses of my daily life working with statistical software in words written more than 70 years ago. Bear with me because this blog post takes the scenic route to arrive at modern statistics. [Read more…] about Statistics, Old Love Letters, and Changing Times
Identifying the Most Important Independent Variables in Regression Models
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
Confidence Intervals vs Prediction Intervals vs Tolerance Intervals
Intervals are estimation methods in statistics that use sample data to produce ranges of values that are likely to contain the population value of interest. In contrast, point estimates are single value estimates of a population value. Of the different types of statistical intervals, confidence intervals are the most well-known. However, certain kinds of analyses and situations call for other types of ranges that provide different information. [Read more…] about Confidence Intervals vs Prediction Intervals vs Tolerance Intervals
Five P Value Tips to Avoid Being Fooled by False Positives and other Misleading Hypothesis Test Results
Despite the popular notion to the contrary, understanding the results of your statistical hypothesis test is not as simple as determining only whether your P value is less than your significance level. In this post, I present additional considerations that help you assess and minimize the possibility of being fooled by false positives and other misleading results. [Read more…] about Five P Value Tips to Avoid Being Fooled by False Positives and other Misleading Hypothesis Test Results
World Travel, Rough Roads, and Manually Adjusting Graph Scales!
As my family and I were being rattled around in a four-wheel drive vehicle in the remote Osa Peninsula in Costa Rica, it struck me that traveling to exotic locations is just like manually adjusting the scales on graphs! That’s probably not what you were expecting, but let me explain! Unlike most of my statistical blog posts, this one gets a bit philosophical! [Read more…] about World Travel, Rough Roads, and Manually Adjusting Graph Scales!
How to Interpret Regression Models that have Significant Variables but a Low R-squared
Does your regression model have a low R-squared? That seems like a problem—but it might not be. Learn what a low R-squared does and does not [glossary_exclude]mean[/glossary_exclude] for your model. [Read more…] about How to Interpret Regression Models that have Significant Variables but a Low R-squared
How High Does R-squared Need to Be?
How high does R-squared need to be in regression analysis? That seems to be an eternal question. [Read more…] about How High Does R-squared Need to Be?
Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes
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
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
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
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 linear regression 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
Nonparametric Tests vs. Parametric Tests
Nonparametric tests don’t require that your data follow the normal distribution. They’re also known as distribution-free tests and can provide benefits in certain situations. Typically, people who perform statistical hypothesis tests are more comfortable with parametric tests than nonparametric tests.
You’ve probably heard it’s best to use nonparametric tests if your data are not normally distributed—or something along these lines. That seems like an easy way to choose, but there’s more to the decision than that. [Read more…] about Nonparametric Tests vs. Parametric Tests
Hypothesis Testing and Confidence Intervals
Confidence intervals and hypothesis testing are closely related because both methods use the same underlying methodology. Additionally, there is a close connection between significance levels and confidence levels. Indeed, there is such a strong link between them that hypothesis tests and the corresponding confidence intervals always agree about statistical significance.
A confidence interval is calculated from a sample and provides a range of values that likely contains the unknown value of a population parameter. To learn more about confidence intervals in general, how to interpret them, and how to calculate them, read my post about Understanding Confidence Intervals.
In this post, I demonstrate how confidence intervals work using graphs and concepts instead of formulas. In the process, I compare and contrast significance and confidence levels. You’ll learn how confidence intervals are similar to significance levels in hypothesis testing. You can even use confidence intervals to determine statistical significance.
Read the companion post for this one: How Hypothesis Tests Work: Significance Levels (Alpha) and P-values. In that post, I use the same graphical approach to illustrate why we need hypothesis tests, how significance levels and P-values can determine whether a result is statistically significant, and what that actually means.
Significance Level vs. Confidence Level
Let’s delve into how confidence intervals incorporate the margin of error. Like the previous post, I’ll use the same type of sampling distribution that showed us how hypothesis tests work. This sampling distribution is based on the t-distribution, our sample size, and the variability in our sample. Download the CSV data file: FuelsCosts.
There are two critical differences between the sampling distribution graphs for significance levels and confidence intervals–the value that the distribution centers on and the portion we shade.
The significance level chart centers on the null value, and we shade the outside 5% of the distribution.
Conversely, the confidence interval graph centers on the sample mean, and we shade the center 95% of the distribution.
The shaded range of sample means [267 394] covers 95% of this sampling distribution. This range is the 95% confidence interval for our sample data. We can be 95% confident that the population mean for fuel costs fall between 267 and 394.
Confidence Intervals and the Inherent Uncertainty of Using Sample Data
The graph emphasizes the role of uncertainty around the point estimate. This graph centers on our sample mean. If the population mean equals our sample mean, random samples from this population (N=25) will fall within this range 95% of the time.
We don’t know whether our sample mean is near the population mean. However, we know that the sample mean is an unbiased estimate of the population mean. An unbiased estimate does not tend to be too high or too low. It’s correct on average. Confidence intervals are correct on average because they use sample estimates that are correct on average. Given what we know, the sample mean is the most likely value for the population mean.
Given the sampling distribution, it would not be unusual for other random samples drawn from the same population to have means that fall within the shaded area. In other words, given that we did, in fact, obtain the sample mean of 330.6, it would not be surprising to get other sample means within the shaded range.
If these other sample means would not be unusual, we must conclude that these other values are also plausible candidates for the population mean. There is inherent uncertainty when using sample data to make inferences about the entire population. Confidence intervals help gauge the degree of uncertainty, also known as the margin of error.
Related post: Sampling Distributions
Confidence Intervals and Statistical Significance
If you want to determine whether your hypothesis test results are statistically significant, you can use either P-values with significance levels or confidence intervals. These two approaches always agree.
The relationship between the confidence level and the significance level for a hypothesis test is as follows:
Confidence level = 1 – Significance level (alpha)
For example, if your significance level is 0.05, the equivalent confidence level is 95%.
Both of the following conditions represent statistically significant results:
- The P-value in a hypothesis test is smaller than the significance level.
- The confidence interval excludes the null hypothesis value.
Further, it is always true that when the P-value is less than your significance level, the interval excludes the value of the null hypothesis.
In the fuel cost example, our hypothesis test results are statistically significant because the P-value (0.03112) is less than the significance level (0.05). Likewise, the 95% confidence interval [267 394] excludes the null hypotheses value (260). Using either method, we draw the same conclusion.
Hypothesis Testing and Confidence Intervals Always Agree
The hypothesis testing and confidence interval results always agree. To understand the basis of this agreement, remember how confidence levels and significance levels function:
- A confidence level determines the distance between the sample mean and the confidence limits.
- A significance level determines the distance between the null hypothesis value and the critical regions.
Both of these concepts specify a distance from the mean to a limit. Surprise! These distances are precisely the same length.
A 1-sample t-test calculates this distance as follows:
The critical t-value * standard error of the mean
Interpreting these statistics goes beyond the scope of this article. But, using this equation, the distance for our fuel cost example is $63.57.
P-value and significance level approach: If the sample mean is more than $63.57 from the null hypothesis mean, the sample mean falls within the critical region, and the difference is statistically significant.
Confidence interval approach: If the null hypothesis mean is more than $63.57 from the sample mean, the interval does not contain this value, and the difference is statistically significant.
Of course, they always agree!
The two approaches always agree as long as the same hypothesis test generates the P-values and confidence intervals and uses equivalent confidence levels and significance levels.
Related posts: Standard Error of the Mean and Critical Values
I Really Like Confidence Intervals!
In statistics, analysts often emphasize using hypothesis tests to determine statistical significance. Unfortunately, a statistically significant effect might not always be practically meaningful. For example, a significant effect can be too small to be important in the real world. Confidence intervals help you navigate this issue!
Similarly, the margin of error in a survey tells you how near you can expect the survey results to be to the correct population value.
Learn more about this distinction in my post about Practical vs. Statistical Significance.
Learn how to use confidence intervals to compare group means!
Finally, learn about bootstrapping in statistics to see an alternative to traditional confidence intervals that do not use probability distributions and test statistics. In that post, I create bootstrapped confidence intervals.
Reference
Neyman, J. (1937). Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability. Philosophical Transactions of the Royal Society A. 236 (767): 333–380.
R-squared Is Not Valid for Nonlinear Regression
Nonlinear regression is an extremely flexible analysis that can fit most any curve that is present in your data. R-squared seems like a very intuitive way to assess the goodness-of-fit for a regression model. Unfortunately, the two just don’t go together. R-squared is invalid for nonlinear regression. [Read more…] about R-squared Is Not Valid for Nonlinear Regression
How t-Tests Work: t-Values, t-Distributions, and Probabilities
T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples. In this post, I show you how t-tests use t-values and t-distributions to calculate probabilities and test hypotheses.
As usual, I’ll provide clear explanations of t-values and t-distributions using concepts and graphs rather than formulas! If you need a primer on the basics, read my hypothesis testing overview. [Read more…] about How t-Tests Work: t-Values, t-Distributions, and Probabilities
How to Interpret the Constant (Y Intercept) in Regression Analysis
The constant term in regression analysis is the value at which the regression line crosses the y-axis. The constant is also known as the y-intercept. That sounds simple enough, right? Mathematically, the regression constant really is that simple. However, the difficulties begin when you try to interpret the meaning of the y-intercept in your regression output. [Read more…] about How to Interpret the Constant (Y Intercept) in Regression Analysis
How F-tests work in Analysis of Variance (ANOVA)
Analysis of variance (ANOVA) uses F-tests to statistically assess the equality of means when you have three or more groups. In this post, I’ll answer several common questions about the F-test.
- How do F-tests work?
- Why do we analyze variances to test means?
I’ll use concepts and graphs to answer these questions about F-tests in the context of a one-way ANOVA example. I’ll use the same approach that I use to explain how t-tests work. If you need a primer on the basics, read my hypothesis testing overview.
To learn more about ANOVA tests, including the more complex forms, read my ANOVA Overview and One-Way ANOVA Overview and Example.
[Read more…] about How F-tests work in Analysis of Variance (ANOVA)
Check Your Residual Plots to Ensure Trustworthy Regression Results!
Use residual plots to check the assumptions of an OLS linear regression model. If you violate the assumptions, you risk producing results that you can’t trust. Residual plots display the residual values on the y-axis and fitted values, or another variable, on the x-axis. After you fit a regression model, it is crucial to check the residual plots. If your plots display unwanted patterns, you can’t trust the regression coefficients and other numeric results.
In this post, I explain the conceptual reasons why residual plots help ensure that your regression model is valid. I’ll also show you what to look for and how to fix the problems. [Read more…] about Check Your Residual Plots to Ensure Trustworthy Regression Results!