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How to Report Pearson’s r in APA Format (With Examples)

A Pearson Correlation Coefficient , often denoted r , measures the linear association between two variables.

It always takes on a value between -1 and 1 where:

• -1 indicates a perfectly negative linear correlation between two variables
• 0 indicates no linear correlation between two variables
• 1 indicates a perfectly positive linear correlation between two variables

We use the following general structure to report a Pearson’s r in APA format:

A Pearson correlation coefficient was computed to assess the linear relationship between [variable 1] and [variable 2] .   There was a [negative or positive] correlation between the two variables, r( df ) = [r value] , p = [p-value] .

Keep in mind the following when reporting Pearson’s r  in APA format:

• Round the p-value to three decimal places.
• Round the value for r to two decimal places.
• Drop the leading 0 for the p-value and r (e.g. use .77, not 0.77)
• The degrees of freedom (df) is calculated as N – 2.

The following examples show how to report Pearson’s r in APA format in various scenarios.

Example 1: Hours Studied vs. Exam Score Received

A professor collected data for the number of hours studied and the exam score received for 40 students in his class. He found the Pearson correlation coefficient between the two variables to be 0.48 with a corresponding p-value of 0.002.

Here is how to report Pearson’s r in APA format:

A Pearson correlation coefficient was computed to assess the linear relationship between hours studied and exam score.   There was a positive correlation between the two variables, r(38) = .48, p = .002.

Example 2: Time Spent Running vs. Body Fat

A doctor collected data for the number of hours spent running per week and body fat percentage for 35 patients. He found the Pearson correlation coefficient between the two variables to be -0.37 with a corresponding p-value of 0.029.

A Pearson correlation coefficient was computed to assess the linear relationship between hours spent running and body fat percentage.   There was a negative correlation between the two variables, r(33) = -.37, p = .029.

Example 3: Ad Spend vs. Revenue Generated

A company collected data for the amount of money spent on advertising and the total revenue generated during 15 consecutive sales periods. They found the Pearson correlation coefficient between the two variables to be 0.71 with a corresponding p-value of 0.003.

A Pearson correlation coefficient was computed to assess the linear relationship between advertising spend and total revenue.   There was a positive correlation between the two variables, r(13) = .71, p = .003.

Additional Resources

The following tutorials explain how to report other statistical tests and procedures in APA format:

How to Report Cronbach’s Alpha (With Examples) How to Report t-Test Results (With Examples) How to Report Regression Results (With Examples) How to Report ANOVA Results (With Examples)

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How to Report Pearson's r (Pearson's Correlation Coefficient) in APA Style

The APA has precise requirements for reporting the results of statistical tests, which means as well as getting the basic format right, you need to pay attention to the placing of brackets, punctuation, italics, and so on.

Happily, the basic format for citing Pearson's r is not too complex, as you can see here (the color red means you substitute in the appropriate value from your study).

r ( degress of freedom ) = the r statistic , p = p value .

Imagine we have conducted a study of 40 students that looked at whether IQ scores and GPA are correlated. We might report the results like this:

IQ and GPA were found to be moderately positively correlated, r (38) = .34, p = .032.

Other Examples

The variables shoe size and height were found to be strongly correlated, r (128) = .89, p < .01.

Among the students of Hogwarts University, the number of hours playing Fortnite per week and midterm exam results were negatively correlated, r (78) = -.45, p < .001.

Here are some things you should watch out for.

1. There are two ways to report p values. The first way is to cite the alpha value as in the second example above. The second way, very much the preferred way in the age of computer aided calculations (and the way recommended by the APA), is to report the exact p value (as in our main example). If you report the exact p value, then you need to state your alpha level early in your results section. The other thing to note here is that if your p value is less than .001, it's conventional simply to state p < .001, rather than give the exact value.

2. The r statistic should be stated at 2 decimal places.

3. Remember to drop the leading 0 from both r and the p value (i.e., not 0.34, but rather .34).

4. You don't need to provide the formula for r .

5. Degrees of freedom for r is N - 2 (the number of data points minus 2).

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Report Correlation in APA Style using R: Text & Tables

In this post, you will learn how to report correlation according to APA. Adhering to APA (American Psychological Association) guidelines is crucial when reporting correlation analysis in academic research. Whether you are conducting research in psychology, cognitive hearing science, or cognitive science, APA style is often required by journals and conferences. This post will provide a step-by-step guide on reporting correlation in APA style using R, including creating tables.

To report correlation in APA style, we can follow a specific template that includes the, e.g. correlation method, sample size, degrees of freedom, correlation coefficient value, and p-value. APA 7th edition also recommends reporting confidence intervals for correlation coefficients. In R, we can use functions such as cor.test( ) to obtain these statistics for Pearson’s r and other correlation methods. Here is how we can report Pearson’s product-moment correlation coefficient according to APA:

Table of Contents

R packages for reporting correlation results, requirements.

• Template for Reporting Pearson’s Correlation (r) in APA Style:

Template for Reporting Spearman’s Rho in APA Style:

Template for reporting kendall’s tau in apa style:, report pearson’s correlation apa, reporting spearman’s correlation apa 7 style with r, kendall’s tau correlation results apa style, apa format correlation table using apatables, 1: load the packages, 2: create a correlation matrix in r, 3: calculate and add mean and standard deviation, 4: create the apa table in r, apatables vs. rempsyc, conclusion: reporting correlation results in apa style with r.

The outline of this post is to provide you with a comprehensive guide to reporting correlation results in APA style using R. Before diving into the specifics, we will discuss the requirements, including installing several R packages. Next, we’ll explore the necessary data and how to report correlation results for Pearson’s correlation (r), Spearman’s Rho, and Kendall’s Tau. Furthermore, we will provide templates for each reporting style and explain how to create a correlation matrix in R.

In the following section, we will guide you through reporting Pearson’s, Spearman’s, and Kendall’s correlation coefficients in APA 7 style using R. We will also show you how to create a correlation table in APA format using the apaTables and rempsyc packages. We will also walk you through the four steps to create an APA formatted table in R: loading the packages, creating a correlation matrix, calculating and adding mean and standard deviation, and creating the table itself. By the end of the post, you’ll have the tools and knowledge to report your correlation results in APA style confidently. Finally, we will compare apaTables and rempsyc for creating APA tables, highlighting some of the pros and cons of each.

There are several R packages available that can help researchers report their results in APA style. One such package is the apaTables package, which provides functions for creating APA-style tables of descriptive statistics, ANOVA results, and correlation matrices. In this post, we will use the apaTables and rempsyc packages. We will also use the report package. This package is also useful for creating APA-style reports, including tables, figures, and statistical analyses. Note the report package is part of the easystats packages, which is a set of very helpful packages (e.g., correlation). Another useful package is papaja , which allows researchers to write APA-style manuscripts in RMarkdown format. Other packages that can be helpful for APA-style reporting include ggplot2 and dplyr . With these packages and others like them, researchers can easily generate high-quality reports and manuscripts that meet APA style guidelines.

To follow this blog post, you should understand R programming concepts, such as data manipulation, syntax, and regular expressions. Additionally, you will need to have some packages installed, including tidyverse , report , corrr , apaTables , and rempsyc .

To install the packages, you can use the install.packages() function followed by the package name within quotation marks. For example, to install tidyverse , you can type install.packages("tidyverse") into the console.

If you plan to use apaTables , you may only need to install that package, as it includes many other necessary packages. However, if you plan to use rempsyc , you must install all the required packages separately. Note it is also recommended that you update R to the latest version .

In this blog post, we will be using tidyverse packages such as dplyr , magittr , and tidyr , to manipulate data. We will also be using the corrr package to create a correlation matrix. Finally, we will use the rempsyc package to create an APA-formatted table of the correlation results.

Here is an example dataset we can use practicing reporting and creating correlation tables following APA 7 style in R:

In the code chunk above, we generated two correlation matrices using the matrix function. The first matrix was for the three Span variables, and the second was for the three Effect variables. We then used the mvrnorm function from the mvtnorm package to generate 100 observations for each of the variables. We assigned these observations to separate data frames using as.data.frame and then used rename from the dplyr package to give them meaningful names.

To create the correlation between the Working Memory Capacity and Inhibition variables, we used the mutate function from dplyr to modify the Working Memory Capacity variables. We multiplied each variable by 0.75 and added 0.25 times the corresponding Effect variable. This gave us Span variables that were moderately correlated with the Effect variables, with correlation coefficients ranging from 0.14 to 0.31.

In the code chunk, we used the MASS and dplyr packages to generate and manipulate data for six variables with different correlation structures, allowing for the simulation of complex relationships between variables commonly used in cognitive psychology research. Here is a quick overview of the data:

Here we get a quick view of how the variables are correlated:

In the next section, we will provide a detailed step-by-step guide on obtaining and reporting correlation statistics in APA style using R, along with examples of how to customize the output to meet specific requirements.

How to Report Correlation Results in APA 7

This section will provide text templates for reporting Pearson’s R, Spearman’s Rho, and Kendall’s Tau in APA style. These templates can be used to report the results of correlation analyses in research articles, theses, and dissertations.

Template for Reporting Pearson’s Correlation ( r ) in APA Style:

Here is a template you can use to report Pearson’s correlation according to APA 7:

Pearson’s correlation was used to assess the relationship between [Variable X] and [Variable Y] (r = [correlation coefficient], p < [p-value], 95% CI [lower bound, upper bound], N = [sample size]).

Here is a template you can use to report Spearman’s correlation according to APA:

Spearman’s rho was used to assess the relationship between [Variable X] and [Variable Y] ( ρ = [correlation coefficient], N = [sample size], p < [p-value].

Here is how you can report Kendall’s Tau using APA style:

Kendall’s tau-b was used to assess the relationship between [Variable X] and [Variable Y] ( τ_b = [correlation coefficient], p < [p-value].

In the next section, we will look at how we can use the R package report for reporting correlation results according to APA format.

How to Report Correlation Results in APA Style in R

Here is how to calculate Pearson’s product-moment correlation coefficient in R and use the report() function:

In the code chunk above, we used the library function to load the report and magrittr packages into our R environment.

Next, we used the %$% operator from the magrittr package to avoid repeatedly referring to the dataframe we are working with. Then, we used the cor.test function to calculate the Pearson’s correlation coefficient between the Stroop and Flanker variables in our data frame.

Finally, we passed the output of cor.test to the report function to generate a report of the correlation analysis. Here is the output:

As you can see, it is not how we should report correlation according to APA. We need to adjust the output from the report() function:

In the code chunk above, the report generated using results.r is stored in report_output variable. Then, the summary() function removes the first sentence from the report. Finally, gsub() function is used to italicize the statistical letters, which are identified using the regular expression \\b(r|CI|t|p)\\b . Note that “|” means “or” in R .

As you can see, it does not entirely follow our previous templates, but it will do.

Here is another example of how we can use R to report correlation results In APA style:

In the code chunk above, gsub() is used to modify the report text generated from report() function. We use gsub() to replace “rho =” with “ r s =” in the report_output variable. Next, we remove “, S = [0-9]+\.[0-9]+” using regex from the report_output . The third gsub() replaces the closing parenthesis in report_output with “, n = {sample size})” using n extracted from the data. We use gsub() a fourth time to italicize the letter “p” in report_output . Finally, the fifth gsub() swaps the places of “ n = {sample size}” and “ p {comparison} {value}” in report_output using regex.

Here is a third example. Here we report Kendall’s Tau according to APA style:

In the code chunk above, we compute Kendall’s tau correlation coefficient. Using gsub() , we first replace “tau =” with “ r τ =” in report_output. Then, we remove “, z =” and the corresponding value from report_output using another gsub() . Lastly, we italicize “p” using a third gsub() . All of these operations are done using the pipe operator. Here is the generated text that follows reporting correlation (Kendall’s tau) in APA style:

Now that we know how to report correlations according to APA 7 using R, we can also use RMarkdown to create APA tables. To create these tables, we need to generate a correlation matrix. This will allow us to easily view all the correlations in our dataset and report them clearly and concisely. Using R and RMarkdown, we can streamline the process of creating APA tables and ensure that they meet the necessary formatting guidelines.

Here is how to get apa correlation table in R:

In the code chunk above, we use the apaTables package to create an APA-style correlation table.

The apa.cor.table() function takes in our data object and generates a correlation matrix according to APA 7 guidelines.

We specify the filename argument to save the table in a Word document named “APA_Correlation_Table.doc”. Additionally, we specify the table.number argument to set the table number as “Table 1”.

That was simple. These few lines of code will create an APA 7-compliant correlation table in R and save it to a Word document for reporting and analysis purposes. Here is the output:

In addition, it is worth mentioning that apaTables is not only capable of creating APA-style correlation tables but also ANOVA tables, regression tables, and more. However, a drawback of using apaTables is that the output can only be saved as Word files and cannot be easily included in PDF or HTML reports. Therefore, we will look at how to use the rempsyc package to report correlation results in an APA formatted table.

How to Report Correlation Results in a APA Formatted Table in R

Here are the steps for creating an APA 7 formatted correlation matrix using the nice_table() function from the rempsyc package:

First, we need to load the corrr and rempsyc packages:

Next, we will create the correlation matrix using correlate() :

In the code chunk above, we are computing a correlation matrix using the correlate() function from the corrr package. Next, we remove the first row and last column of the correlation matrix using slice() and select() functions. Finally, we remove the upper triangle of the matrix and replace those values with NAs using shave() . Note how we used the dplyr’s select() to remove a column in R . In the following step, we will calculate and add the mean and the standard deviation to the dataframe.

The third step is to calculate and add mean and standard deviation to the correlation matrix:

In the code chunk above, we first calculate the mean and standard deviation of the dataset. Next, we transform the data from wide to long format using pivot_longer() . We separate the variable names into two columns: “term” and “Stat”. To maintain the order of the variables in the long format, we change the “term” column to a factor with unique levels. Then, we spread the data so that we have two columns: “mean” and “sd”. After that, we add the correlation matrix to the table using left_join() and by = "term" . Finally, we remove the last column of the table using select(-last_col()) and round.

Here is how to use the nice_table() function to generate an APA formatted correlation table in R:

In the code chunk above, we create an APA-formatted correlation table using the rempsyc package’s nice_table function. First, we rename a column to “Variable” with rename() . Then, we use nice_table() to create the table. We set the second and third columns (mean and standard deviation) to be in italics using italics = 2:3 . We add a title and a note with the title and note arguments, respectively. The note specifies that “M” and “SD” are used to represent mean and standard deviation.

We can also save the APA formatted table in R a .docx file:

Here are the correlation results in a table formatted according to APA 7:

As you may have noticed, we had to write more code than when using the apaTables package.

apaTables is a straightforward package with a single function for creating APA-style tables. It allows for quick and easy creation of tables and is useful for those who do not want to spend much time customizing their tables. However, a downside of apaTables is that it only works for creating Word documents, not PDFs.

In contrast, rempsyc requires more coding but provides many table customization options. rempsyc can create tables in both Word and PDF formats, making it a more versatile option. Additionally, rempsyc includes several functions for creating tables with descriptive statistics, regression results, and more. However, as you have seen above, we would have to calculate the confidence interval when using rempsyc .

In summary, apaTables is a good option for those who want a quick and easy solution for creating simple APA-style tables in Word documents. However, if you need more customization options or want to create tables in PDF format, rempsyc might be a better choice despite requiring more coding.

This blog post covered different R packages and templates for reporting correlation results in APA style. It also explained the requirements for following the post, including basic R knowledge and installing necessary packages. The post demonstrated how to report Pearson’s r, Spearman’s rho, and Kendall’s tau correlation results. It also showed how to create an APA-formatted table using the apaTables package and customize a table using rempsyc . The post outlined the steps for creating a correlation matrix, adding mean and standard deviation, and creating an APA table.

While apaTables offers a simple solution for creating an APA-formatted table, it cannot be saved as a PDF. In contrast, rempsyc provides more customization options but requires more coding.

To conclude, this blog post provided a comprehensive guide for reporting correlation results in APA style using R. If you found this post helpful, please share it on social media and leave a comment to let us know your thoughts.

Here are other resources you might find helpful:

• How to Convert a List to a Dataframe in R – dplyr
• Papaja – APA manuscripts made easy
• R Count the Number of Occurrences in a Column using dplyr
• Countif function in R with Base and dplyr
• How to Make a Residual Plot in R & Interpret Them using ggplot2
• Plot Prediction Interval in R using ggplot2
• How to Standardize Data in R
• Sum Across Columns in R – dplyr & base
• How to use %in% in R: 8 Example Uses of the Operator
• How to Calculate Z Score in R

3 thoughts on “Report Correlation in APA Style using R: Text & Tables”

In your code, you load dplyr before MASS, so there is a namespace conflict for `select()` later on. It would be better to load MASS, then dplyr, else users might not know why the code is not working for them 🙂

“` r library(dplyr) library(MASS) #> Attaching package: ‘MASS’ #> The following object is masked from ‘package:dplyr’: #> #> select

# Now we add the correlation matrix corr_tab select(-last_col()) #> Error in select(mtcars, -last_col()): unused argument (-last_col()) “`

Created on 2023-05-09 with [reprex v2.0.2]( https://reprex.tidyverse.org )

Thank you for your comment. You are right. I forgot to flip that after noticing that error myself. I have now updated the code chunk. By the way, thanks for a great package (or packages, I should say. I also use lavaanExtra).

You are welcome! Thanks for featuring my work and for this thorough comparison on generating formatted correlation tables. It will be a good reference for everyone looking into this matter.

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How to Report Results of Multiple Linear Regression in APA Style

You will learn How to Report Results of Multiple Linear Regression , accurately reporting coefficients, significance levels, and assumptions using APA style.

Introduction

Multiple linear regression is a fundamental statistical method to understand the relationship between one dependent variable and two or more independent variables. This approach allows researchers and analysts to predict the dependent variable’s outcome based on the independent variables’ values, providing insights into complex relationships within data sets. The power of multiple linear regression lies in its ability to control for various confounding factors simultaneously, making it an invaluable tool in fields ranging from social sciences to finance and health sciences.

Reporting the results of multiple linear regression analyses requires precision and adherence to established guidelines, such as those provided by the American Psychological Association (APA) style. The importance of reporting in APA style cannot be overstated, as it ensures clarity, uniformity, and comprehensiveness in research documentation. Proper reporting includes:

• Detailed information about the regression model used.
• The significance of the predictors.
• The fit of the model.
• Any assumptions or conditions that were tested.

Adhering to APA style enhances the readability and credibility of research findings, facilitating their interpretation and application by a broad audience.

This guide will equip you with the knowledge and skills to effectively report multiple linear regression results in APA style, ensuring your research communicates scientific inquiries.

• Detail assumptions check like multicollinearity with VIF scores.
• Report the adjusted R-squared to express model fit.
• Identify significant predictors with t-values and p-values in your regression model.
• Include confidence intervals for a comprehensive understanding of predictor estimates.
• Explain model diagnostics with residual plots for validity.

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Step-by-Step Guide with Examples

1. objective of regression analysis.

Initiate by clearly stating the purpose of your multiple linear regression (MLR) analysis. For example, you might explore how environmental factors (X1, X2, X3) predict plant growth (Y).  Example:   “This study aims to assess the impact of sunlight exposure (X1), water availability (X2), and soil quality (X3) on plant growth rate (Y).”

2. Sample Size and Power

Discuss the significance of your sample size. A larger sample provides greater power for a robust MLR analysis.  Example:  “With a sample size of 200 plants, we ensure sufficient power to detect significant predictors of growth, minimizing type II errors.”

*Considering the importance of the power of the statistical test, calculating the sample size is a crucial step for accurately determining the adequate sample size needed to identify the estimated relationship.

3. Checking and Reporting Model Assumptions

• Linearity : Verify each independent variable’s relationship with the dependent variable is linear.  Example:   “Scatterplots of sunlight exposure, water availability, and soil quality against plant growth revealed linear trends.”
• Normality of Residuals : Assess using the Shapiro-Wilk test.  Example:   “The Shapiro-Wilk test confirmed the residuals’ normality, W = .98, p = .15.”
• Homoscedasticity : Evaluate with the Breusch-Pagan test.  Example:   “Homoscedasticity was confirmed, with a Breusch-Pagan test result of χ² = 5.42, p = 0.14.”
• Independence of Errors : Use the Durbin-Watson statistic.  Example:   “The Durbin-Watson statistic of 1.92 suggests no autocorrelation, indicating independent errors.”

4. Statistical Significance of the Regression Model

Present the F-statistic, degrees of freedom, and its significance (p-value) to demonstrate the model’s overall fit.  Example:   “The model was significant, F(3,196) = 12.57, p < 0.001, indicating at least one predictor significantly affects plant growth.”

5. Coefficient of Determination

Report the adjusted R² to show the variance explained by the model.  Example:   “The model explains 62% of the variance in plant growth, with an adjusted R² of 0.62.”

6. Statistical Significance of Predictors

Detail each predictor’s significance through t-tests.  Example:   “Sunlight exposure was a significant predictor, t(196) = 5.33, p < 0.001, indicating a positive effect on plant growth.”

7. Regression Coefficients and Equation

Provide the regression equation with unstandardized coefficients.  Example:   “The regression equation was Y = 2.5 + 0.8X1 + 0.5X2 – 0.2X3, where each hour of sunlight (X1) increases growth by 0.8 units…”

8. Discussion of Model Fit and Limitations

Reflect on how well the model fits the data and its limitations.  Example:   “While the model fits well (Adjusted R² = 0.62), it’s crucial to note that it does not prove causation, and external factors not included in the model may also affect plant growth.”

9. Additional Diagnostics and Visualizations

Incorporate diagnostics like VIF for multicollinearity and visual aids.  Example:   “VIF scores were below 5 for all predictors, indicating no multicollinearity concern. Residual plots showed random dispersion, affirming model assumptions.”

“In our exploration of the determinants of final exam scores in a university setting, we employed a multiple linear regression model to assess the contributions of study hours (X1), class attendance (X2), and student motivation (X3). The model, specified as Y = β0 + β1X1 + β2X2 + β3X3 + ε, where Y represents final exam scores, aimed to provide a comprehensive understanding of how these variables collectively influence academic performance.

Assumptions Check:  Before examining the predictive power of our model, a thorough assessment of its foundational assumptions was undertaken to affirm the integrity of our analysis. Scatterplot examinations scrutinized each predictor’s relationship with the dependent variable for linearity, revealing no deviations from linear expectations. The Shapiro-Wilk test substantiated the normality of the residuals (W = .98, p = .15), thereby satisfying the normality criterion. Homoscedasticity, the uniform variance of residuals across the range of predicted values, was confirmed via the Breusch-Pagan test (χ² = 5.42, p = 0.14). Furthermore, the Durbin-Watson statistic stood at 1.92, effectively ruling out autocorrelation among residuals and attesting to the independence of errors. The Variance Inflation Factor (VIF) for each predictor was well below the threshold of 5, dispelling multicollinearity concerns. Collectively, these diagnostic tests validated the key assumptions underpinning our multiple linear regression model, providing a solid groundwork for the subsequent analysis.

Model Summary : The overall fit of the model was statistically significant, as indicated by an F-statistic of 53.24 with a p-value less than .001 (F(3,196) = 53.24, p < .001), suggesting that the model explains a significant portion of the variance in exam scores. The adjusted R² value of .43 further illustrates that our model can account for approximately 43% of the final exam scores variability, highlighting the included predictors’ substantial impact.

Coefficients and Confidence Intervals :

• The intercept, β0, was estimated at 50 points, implying an average exam score baseline when all independent variables are held at zero.
• Study Hours (X1) : Each additional hour of study was associated with a 2.5 point increase in exam scores (β1 = 2.5), with a 95% confidence interval of [1.9, 3.1], underscoring the value of dedicated study time.
• Class Attendance (X2) : Regular attendance contributed an additional 1.8 points to exam scores per class attended (β2 = 1.8), with the confidence interval ranging from 1.1 to 2.5, reinforcing the importance of class participation.
• Student Motivation (X3) : Motivation emerged as a significant factor, with a 3.2-point increase in scores for heightened motivation levels (β3 = 3.2) and a confidence interval of [2.4, 4.0], suggesting a profound influence on academic success.

Model Diagnostics : The diagnostic checks, including the analysis of residuals, confirmed the model’s adherence to the assumptions of linear regression. The absence of discernible patterns in the residual plots affirmed the model’s homoscedasticity and linearity, further solidifying the reliability of our findings.

In conclusion, our regression analysis elucidates the critical roles of study hours, class attendance, and student motivation in determining final exam scores. The robustness of the model, evidenced by the stringent checks and the significant predictive power of the included variables, provides compelling insights into effective academic strategies. These findings validate our initial hypotheses and offer valuable guidance for educational interventions to enhance student outcomes.

These results, particularly the point estimates and their associated confidence intervals, provide robust evidence supporting the hypothesis that study hours, class attendance, and student motivation are significant predictors of final exam scores. The confidence intervals offer a range of plausible values for the true effects of these predictors, reinforcing the reliability of the estimates.

In this comprehensive guide, we’ve navigated the intricacies of reporting multiple linear regression results in APA style, emphasizing the critical components that must be included to ensure clarity, accuracy, and adherence to standardized reporting conventions. Key points such as the importance of presenting a clear model specification, conducting thorough assumption checks, detailing model summaries and coefficients, and interpreting the significance of predictors have been highlighted to assist you in crafting a report that stands up to academic scrutiny and contributes valuable insights to your field of study.

Accurate reporting is paramount in scientific research. It conveys findings and upholds the integrity and reproducibility of the research process. By meticulously detailing each aspect of your multiple linear regression analysis, from the initial model introduction to the final diagnostic checks, you provide a roadmap for readers to understand and potentially replicate your study. This level of transparency is crucial for fostering trust in your conclusions and encouraging further exploration and discussion within the scientific community.

Moreover, the practical example is a template for effectively applying these guidelines, illustrating how theoretical principles translate into practice. By following the steps outlined in this guide, researchers can enhance the impact and reach of their studies, ensuring that their contributions to knowledge are recognized, understood, and built upon.

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APA Style T-Test Reporting Guide

Frequently asked questions (faqs).

Multiple linear regression extends simple linear regression by incorporating two or more predictors to explain the variance in a dependent variable, offering a more comprehensive analysis of complex relationships.

Use multiple linear regression to understand the impact of several independent variables on a single outcome and when these variables are expected to interact with each other in influencing the dependent variable.

Key steps include testing for linearity, examining residual plots for homoscedasticity and normality, checking VIF scores for multicollinearity, and using the Durbin-Watson statistic to assess the independence of residuals.

Coefficients represent the expected change in the dependent variable for a one-unit change in the predictor, holding all other predictors constant. Positive coefficients indicate a direct relationship, while negative coefficients suggest an inverse relationship.

Adjusted R-squared provides a more accurate measure of the model’s explanatory power by adjusting for the number of predictors, preventing overestimating variance explained in models with multiple predictors.

Confidence intervals offer a range of plausible values for each coefficient, providing insights into the precision of the estimates and the statistical significance of predictors.

Consider combining highly correlated variables, removing some, or using techniques like principal component analysis to reduce multicollinearity without losing critical information.

Residual analysis can reveal patterns that suggest violations of linear regression assumptions, guiding modifications to the model, such as transforming variables or adding interaction terms.

P-values can be misleading in the presence of multicollinearity, when sample sizes are very large or small, or when data do not meet the assumptions of linear regression, emphasizing the importance of comprehensive diagnostic checks.

Ensure your graphs are clear, labeled accurately, and include necessary details like confidence intervals or regression lines. Follow APA guidelines for figure presentation to maintain consistency and readability in your report.

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Reporting Statistics in APA Style

Dr. jeffrey kahn, illinois state university.

The sample as a whole was relatively young ( M = 19.22, SD = 3.45). The average age of students was 19.22 years ( SD = 3.45).

Nearly half (49%) of the sample was married.

The percentage of participants that were married did not differ by gender, c 2 (1, N = 90) = 0.89, p > .05.

There was a significant effect for gender, t (54) = 5.43, p < .001, with men receiving higher scores than women.

There was a significant main effect for treatment, F (1, 145) = 5.43, p < .01, and a significant interaction, F (2, 145) = 3.13, p < .05.

The two variables were strongly correlated, r (55) = .49, p < .01.

Social support significantly predicted depression scores, b = -.34, t (225) = 6.53, p < .01. Social support also explained a significant proportion of variance in depression scores, R 2 = .12, F (1, 225) = 42.64, p < .01.

Format Outputs of Statistical Tests According to APA Guidelines

Documentation for package ‘apa’ version 0.3.4

• DESCRIPTION file .
• User guides, package vignettes and other documentation.
• Package NEWS .

apaTables Version 2.0

A common task faced by researchers is the creation of APA style (i.e., American Psychological Association style) tables from statistical output. In R a large number of function calls are often needed to obtain all of the desired information for a single APA style table. As well, the process of manually creating APA style tables in a word processor is prone to transcription errors. This package creates Word files (.doc files) containing APA style tables for several types of analyses. Using this package minimizes transcription errors and reduces the number commands needed by the user.

The development version of apaTables R package is hosted here on Github. Current stable version is on the CRAN, see apaTables here.

Install Stable CRAN Version

Install development version.

You can learn how to use apaTables here .

Monthly Downloads

Pull requests, last published, functions in apatables (2.0.8).

“From R to your manuscript”

report ’s primary goal is to bridge the gap between R’s output and the formatted results contained in your manuscript. It automatically produces reports of models and data frames according to best practices guidelines (e.g., APA ’s style), ensuring standardization and quality in results reporting.

Installation

The package is available on CRAN and can be downloaded by running:

If you would instead like to experiment with the development version, you can download it from GitHub :

Load the package every time you start R

Tip Instead of library(report) , use library(easystats) . This will make all features of the easystats-ecosystem available. To stay updated, use easystats::install_latest() .

Documentation

The package documentation can be found here .

Report all the things

General Workflow

The report package works in a two step fashion. First, you create a report object with the report() function. Then, this report object can be displayed either textually (the default output) or as a table, using as.data.frame() . Moreover, you can also access a more digest and compact version of the report using summary() on the report object.

The report() function works on a variety of models, as well as other objects such as dataframes:

These reports nicely work within the tidyverse workflow:

t -tests and correlations

Reports can be used to automatically format tests like t -tests or correlations.

As mentioned, you can also create tables with the as.data.frame() functions, like for example with this correlation test:

This works great with ANOVAs, as it includes effect sizes and their interpretation.

Generalized Linear Models (GLMs)

Reports are also compatible with GLMs, such as this logistic regression :

Mixed Models

Mixed models, whose popularity and usage is exploding, can also be reported:

Bayesian Models

Bayesian models can also be reported using the new SEXIT framework, which combines clarity, precision and usefulness.

Other types of reports

Specific parts.

One can, for complex reports, directly access the pieces of the reports:

Report participants’ details

This can be useful to complete the Participants paragraph of your manuscript.

Report sample

Report can also help you create a sample description table (also referred to as Table 1 ).

Report system and packages

Finally, report includes some functions to help you write the data analysis paragraph about the tools used.

If you like it, you can put a star on this repo, and cite the package as follows:

report is a young package in need of affection . You can easily be a part of the developing community of this open-source software and improve science! Don’t be shy, try to code and submit a pull request (See the contributing guide ). Even if it’s not perfect, we will help you make it great!

Code of Conduct

Please note that the report project is released with a Contributor Code of Conduct . By contributing to this project, you agree to abide by its terms.

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Derrick white joins the garden report after the boston celtics' game 3 win vs. cavs, share this article.

The hosts of the CLNS Media “Garden Report” podcast, Bobby Manning, Josue Pavon, Jimmy Toscano, A. Sherrod Blakely and John Zannis, have long been one of the best postgame pods in the business, and regularly bring quality content to Boston Celtics fans. But it is not all that often that they have one of the players they are podding about stop by the show after the game.

But that is what happened on the latest episode of the “Garden Report,” with star Celtics guard Derrick White stopping by for a few minutes postgame to chat about what was different for Boston in their Game 3 win.

Take a look at the clip embedded below to hear what they had to say.

If you enjoy this pod, check out the “ How Bout Them Celtics ,” “ First to the Floor, ” and the many other New England sports podcasts available on the CLNS Media network.

Listen to the “ Celtics Lab ” podcast on:

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Mass. lawmakers suggest changes to police handling of rape cases after WBUR investigation

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<iframe width="100%" height="124" scrolling="no" frameborder="no" src="https://player.wbur.org/news/2024/05/09/massachusetts-legislature-boston-police-serial-rapist"></iframe>

• Walter Wuthmann

Several state lawmakers say Massachusetts law enforcement needs to more aggressively pursue sexual assault cases. This comes on the heels of a WBUR investigation into how authorities failed to arrest an alleged serial rapist for years, despite DNA evidence.

Legislators say potential improvements could include launching a new statewide sexual assault unit, forcing police to disclose more information about rape reports and expanding the state’s DNA database.

WBUR's investigation found police and prosecutors repeatedly declined to arrest or charge a former Uber driver, Alvin Campbell Jr., despite a string of women reporting assaults from 2016 to 2018, with DNA evidence pointing to Campbell.

Boston police finally arrested Campbell after a fifth victim came forward in 2019 – and even she complained in text messages summarized in court documents that police "were not helpful" initially.

Campbell, the brother of Attorney General Andrea Campbell, is now in jail awaiting trial on charges of raping eight women and attempting to rape a ninth. He has pleaded not guilty on all charges. He has also been linked by DNA evidence to two other rapes, according to a search warrant application . WBUR found 10 of the attacks allegedly occurred after the first woman went to police.

“My initial reaction is just frustration on behalf of all survivors who report their sexual violence to law enforcement,” said Leominster Rep. Natalie Higgins, a Democrat who co-chairs the Caucus of Women Legislators’ task force on sexual violence.

Nationally, only one in five people accused of sexual assault is ever arrested, and even fewer are convicted, according to a study sponsored by the research arm of the U.S. Department of Justice.

Some outside observers questioned whether authorities may have been hesitant to arrest Campbell because he has a prominent sister. Andrea Campbell was a Boston city councilor at the time of her brother's alleged crimes. She told WBUR in a statement she would “never interfere with or influence any investigation, especially one involving such serious allegations.” A spokeswoman said she has had no contact with law enforcement about the investigation or prosecution.

Both Medford police and the Suffolk County District Attorney’s office said the family relationship was not a factor in their response. Boston police declined to answer questions about the story.

Higgins, the state representative, said the case shows that Massachusetts could use a statewide team to help local departments and track down serial offenders who have crossed city or county lines to attack victims. Some other states, like Rhode Island and New Hampshire , already have similar units.

“I think this case really highlights the need for this,” she said.

Higgins said members of the women’s caucus are researching legislation and lobbying Gov. Maura Healey’s administration to take executive action. A spokesperson for Healey said the governor would review “any legislation that reaches her desk.”

WBUR reported police also invoked the state’s sweeping privacy laws to withhold documents about how they handled the investigation, making it harder to know why police took so long to arrest Campbell. Massachusetts is the only state in the country that requires police to withhold all arrests and reports involving sexual and domestic violence.

Some police departments also say the law bars them from telling the public about attacks. Quincy, for instance, said it recorded six unsolved rape cases from 2019 to 2021, but never notified residents and claims it can't reveal details because of the law.

House Republican Leader Brad Jones said he was concerned about “the constraints of the law" in terms of notifying the public and sharing information.

“That stood out to me because that's something that maybe is ripe for change,” he said.

Westfield state Sen. John Velis filed a bill last year that would create a task force to review the statute and make recommendations. The proposal was sent to study in February, effectively killing the bill for the session. Velis said he intends to re-file the bill in the next session, which begins in January.

He noted that other states have found ways to safeguard victims’ privacy, such as redacting their names and other identifying information from documents, instead of withholding everything from the public. “I think that’s the right balance,” Velis said.

Some lawmakers, including Jones and Higgins, also support a bill to ensure the state’s DNA database of convicted offenders is up-to-date and regularly audited. Legislators say the state failed to collect DNA from 10,000 to 15,000 people who are supposed to be in the database. Supporters say expanding the database could make it easier to solve rapes and other cold cases.

In Campbell's case, police already had his DNA on file because he had cycled in and out of prison on past assault and gun charges.

Boston Mayor Michelle Wu declined last month to explain why Boston waited years to arrest Campbell, citing the ongoing criminal proceedings and the state privacy laws.

“Ultimately the law is right now that those documents and that information is shielded from public records,” she said in an interview with WBUR’s Radio Boston .

Boston City Council President Ruthzee Louijeune also declined comment through a spokesperson.

Campbell has not been charged in two of the alleged rapes where DNA identified him as a suspect, including one in Medford in 2018. Medford police did not provide an explanation.

In Medford, Mayor Breanna Lungo-Koehn’s office declined to comment and Medford City Council President Isaac "Zac" Bears did not return messages left with his office.

The state attorney general’s office, which has a new police accountability unit, declined to say whether it planned to examine how police and prosecutors handled the allegations against Alvin Campbell. Executive Office of Public Safety and Security spokesman Tim McGuirk said his state agency does not have statutory authority over municipal police investigations.

The attorney general’s office said Andrea Campbell has recused herself from her brother’s case and that the prosecution has her “unreserved support.” Alvin Campbell’s trial is currently scheduled to begin in December.

Higgins, the state representative from Leominster, said she’d like to think that “all of our departments are doing everything that they can to seek justice for survivors.”

But, she said, “we are met with story after story where that's not the case.”

This segment aired on May 9, 2024.

• We need your help to report on sexual assaults in Massachusetts
• Boston police failed to arrest 'serial rapist' for years despite DNA evidence
• Wu spoke with police leaders after report on alleged serial rapist
• How a Mass. law intended to protect victims became a 'gift to abusers'

Walter Wuthmann State Politics Reporter Walter Wuthmann is a state politics reporter for WBUR.

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Illinois basketball star Terrence Shannon Jr. ordered to stand trial on a rape charge in Kansas

University of Illinois basketball standout Terrence Shannon Jr. appears in court Friday, May 10, 2024, in Lawrence, Kan. A judge ordered him to stand trial on a rape charge. Shannon, 23, of Champaign, Ill., is accused of committing sexual assault on Sept. 9, in Lawrence. (Chris Conde/The Lawrence Journal-World via AP)

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LAWRENCE, Kan. (AP) — A Kansas judge on Friday ordered University of Illinois basketball standout Terrence Shannon Jr. to stand trial on a rape charge following testimony from a woman who said she was “terrified” as she was being assaulted.

Shannon, 23, of Champaign, Illinois, is accused of committing sexual assault on Sept. 9, in Lawrence, Kansas. The Lawrence Journal-World reported that Shannon was in Lawrence that day to watch a football game between the Illini and the University of Kansas.

He has been charged with rape or an alternative count of sexual battery. The charge led to Shannon being suspended for six games before a federal judge reinstated him , ruling his civil rights had been violated.

A woman told police she was at a bar when a man she later identified as Shannon grabbed her buttocks and reached under her skirt and touched her sexually. The woman said the bar was crowded, so she couldn’t move.

Shannon testified on Friday that he never touched the woman, the Journal-World reported.

Judge Sally Pokorny ordered Shannon to stand trial and allowed him to remain free on $50,000 bond.

As a fifth-year student this past season, Shannon led the Illini in scoring and helped them reach the Elite Eight of the NCAA Tournament before losing to eventual champion Connecticut .

Trump May Owe $100 Million From Double-Dip Tax Breaks, Audit Shows

A previously unknown focus of an I.R.S. audit is a dubious accounting maneuver that effectively meant taking the same write-offs twice on a Chicago skyscraper.

The I.R.S. believes that former President Donald J. Trump violated a law meant to prevent double-dipping on tax-reducing losses. Credit... Jamie Kelter Davis for The New York Times

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By Russ Buettner and Paul Kiel

This article was published in partnership with ProPublica. Russ Buettner of The New York Times has spent years reporting on the former president’s finances, including decades of his tax returns. Paul Kiel of ProPublica has reported on the I.R.S. and the ways the ultrawealthy avoid taxes since 2018.

• May 11, 2024

Former President Donald J. Trump used a dubious accounting maneuver to claim improper tax breaks from his troubled Chicago tower, according to an Internal Revenue Service inquiry uncovered by The New York Times and ProPublica. Losing a yearslong audit battle over the claim could mean a tax bill of more than $100 million.

Listen to this article, read by Eric Jason Martin

The 92-story, glass-sheathed skyscraper along the Chicago River is the tallest and, at least for now, the last major construction project by Mr. Trump. Through a combination of cost overruns and the bad luck of opening in the teeth of the Great Recession, it was also a vast money loser.

But when Mr. Trump sought to reap tax benefits from his losses, the I.R.S. has argued, he went too far and in effect wrote off the same losses twice.

The first write-off came on Mr. Trump’s tax return for 2008. With sales lagging far behind projections, he claimed that his investment in the condo-hotel tower met the tax code definition of “worthless,” because his debt on the project meant he would never see a profit. That move resulted in Mr. Trump reporting losses as high as $651 million for the year, The Times and ProPublica found.

There is no indication the I.R.S. challenged that initial claim, though that lack of scrutiny surprised tax experts consulted for this article. But in 2010, Mr. Trump and his tax advisers sought to extract further benefits from the Chicago project, executing a maneuver that would draw years of inquiry from the I.R.S. First, he shifted the company that owned the tower into a new partnership. Because he controlled both companies, it was like moving coins from one pocket to another. Then he used the shift as justification to declare $168 million in additional losses over the next decade.

The issues around Mr. Trump’s case were novel enough that, during his presidency, the I.R.S. undertook a high-level legal review before pursuing it. The Times and ProPublica, in consultation with tax experts, calculated that the revision sought by the I.R.S. would create a new tax bill of more than $100 million, plus interest and potential penalties.

Mr. Trump’s tax records have been a matter of intense speculation since the 2016 presidential campaign, when he defied decades of precedent and refused to release his returns, citing a long-running audit. A first, partial revelation of the substance of the audit came in 2020, when The Times reported that the I.R.S. was disputing a $72.9 million tax refund that Mr. Trump had claimed starting in 2010. That refund, which appeared to be based on Mr. Trump’s reporting of vast losses from his long-failing casinos, equaled every dollar of federal income tax he had paid during his first flush of television riches, from 2005 through 2008, plus interest.

The reporting by The Times and ProPublica about the Chicago tower reveals a second component of Mr. Trump’s quarrel with the I.R.S. This account was pieced together from a collection of public documents, including filings from the New York attorney general’s suit against Mr. Trump in 2022, a passing reference to the audit in a congressional report that same year and an obscure 2019 I.R.S. memorandum that explored the legitimacy of the accounting maneuver. The memorandum did not identify Mr. Trump, but the documents, along with tax records previously obtained by The Times and additional reporting, indicated that the former president was the focus of the inquiry.

It is unclear how the audit battle has progressed since December 2022, when it was mentioned in the congressional report. Audits often drag on for years, and taxpayers have a right to appeal the I.R.S.’s conclusions. The case would typically become public only if Mr. Trump chose to challenge a ruling in court.

In response to questions for this article, Mr. Trump’s son Eric, executive vice president of the Trump Organization, said: “This matter was settled years ago, only to be brought back to life once my father ran for office. We are confident in our position, which is supported by opinion letters from various tax experts, including the former general counsel of the I.R.S.”

An I.R.S. spokesman said federal law prohibited the agency from discussing private taxpayer information.

The outcome of Mr. Trump’s dispute could set a precedent for wealthy people seeking tax benefits from the laws governing partnerships. Those laws are notoriously complex, riddled with uncertainty and under constant assault by lawyers pushing boundaries for their clients. The I.R.S. has inadvertently further invited aggressive positions by rarely auditing partnership tax returns.

The audit represents yet another potential financial threat — albeit a more distant one — for Mr. Trump, the Republicans’ presumptive 2024 presidential nominee. In recent months, he has been ordered to pay $83.3 million in a defamation case and another $454 million in a civil fraud case brought by the New York attorney general, Letitia James. Mr. Trump has appealed both judgments. (He is also in the midst of a criminal trial in Manhattan, where he is accused of covering up a hush-money payment to a porn star in the weeks before the 2016 election.)

Beyond the two episodes under audit, reporting by The Times in recent years has found that, across his business career, Mr. Trump has often used what experts described as highly aggressive — and at times, legally suspect — accounting maneuvers to avoid paying taxes. To the six tax experts consulted for this article, Mr. Trump’s Chicago accounting maneuvers appeared to be questionable and unlikely to withstand scrutiny.

“I think he ripped off the tax system,” said Walter Schwidetzky , a law professor at the University of Baltimore and an expert on partnership taxation.

From ‘$1.2 Billion’ to ‘Worthless’

Mr. Trump struck a deal in 2001 to acquire land and a building that was then home to the Chicago Sun-Times newspaper. Two years later, after publicly toying with the idea of constructing the world’s tallest building there, he unveiled plans for a more modest tower, with 486 residences and 339 “hotel condominiums” that buyers could use for short stays and allow Mr. Trump’s company to rent out. He initially estimated that construction would last until 2007 and cost $650 million.

Mr. Trump placed the project at the center of the first season of “The Apprentice” in 2004 , offering the winner a top job there under his tutelage. “It’ll be a mind-boggling job to manage,” Mr. Trump said during the season finale. “When it’s finished in 2007, the Trump International Hotel and Tower, Chicago, could have a value of $1.2 billion and will raise the standards of architectural excellence throughout the world.”

As his cost estimates increased, Mr. Trump arranged to borrow as much as $770 million for the project — $640 million from Deutsche Bank and $130 million from Fortress Investment Group, a hedge fund and private equity company. He personally guaranteed $40 million of the Deutsche loan. Both Deutsche and Fortress then sold off pieces of the loans to other institutions, spreading the risk and potential gain.

Mr. Trump planned to sell enough of the 825 units to pay off his loans when they came due in May 2008. But when that date came, he had sold only 133. At that point, he projected that construction would not be completed until mid-2009, at a revised cost of $859 million.

He asked his lenders for a six-month extension. A briefing document prepared for the lenders, obtained by The Times and ProPublica, said Mr. Trump would contribute $89 million of his own money, $25 million more than his initial plan. The lenders agreed.

But sales did not pick up that summer, with the nation plunged into the financial crisis that would become the Great Recession. When Mr. Trump asked for another extension in September, his lenders refused.

Two months later, Mr. Trump defaulted on his loans and sued his lenders, characterizing the financial crisis as the kind of catastrophe, like a flood or hurricane, covered by the “force majeure” clause of his loan agreement with Deutsche Bank. That, he said, entitled him to an indefinite delay in repaying his loans. Mr. Trump went so far as to blame the bank and its peers for “creating the current financial crisis.” He demanded $3 billion in damages.

At the time, Mr. Trump had paid down his loans with $99 million in sales but still needed more money to complete construction. At some point that year, he concluded that his investment in the tower was worthless, at least as the term is defined in partnership tax law.

Mr. Trump’s worthlessness claim meant only that his stake in 401 Mezz Venture, the L.L.C. that held the tower, was without value because he expected that sales would never produce enough cash to pay off the mortgages, let alone turn a profit.

When he filed his 2008 tax return, he declared business losses of $697 million. Tax records do not fully show which businesses generated that figure. But working with tax experts, The Times and ProPublica calculated that the Chicago worthlessness deduction could have been as high as $651 million, the value of Mr. Trump’s stake in the partnership — about $94 million he had invested and the $557 million loan balance reported on his tax returns that year.

When business owners report losses greater than their income in any given year, they can retain the leftover negative amount as a credit to reduce their taxable income in future years. As it turned out, that tax-reducing power would be of increasing value to Mr. Trump. While many of his businesses continued to lose money, income from “The Apprentice” and licensing and endorsement agreements poured in: $33.3 million in 2009, $44.6 million in 2010 and $51.3 million in 2011.

Mr. Trump’s advisers girded for a potential audit of the worthlessness deduction from the moment they claimed it, according to the filings from the New York attorney general’s lawsuit. Starting in 2009 Mr. Trump’s team excluded the Chicago tower from the frothy annual “statements of financial condition” that Mr. Trump used to boast of his wealth, out of concern that assigning value to the building would conflict with its declared worthlessness, according to the attorney general’s filing. (Those omissions came even as Mr. Trump fraudulently inflated his net worth to qualify for low-interest loans, according to the ruling in the attorney general’s lawsuit.)

Mr. Trump had good reason to fear an audit of the deduction, according to the tax experts consulted for this article. They believe that Mr. Trump’s tax advisers pushed beyond what was defensible.

The worthlessness deduction serves as a way for a taxpayer to benefit from an expected total loss on an investment long before the final results are known. It occupies a fuzzy and counterintuitive slice of tax law. Three decades ago, a federal appeals court ruled that the judgment of a company’s worthlessness could be based in part on the opinion of its owner. After taking the deduction, the owner can keep the “worthless” company and its assets. Subsequent court decisions have only partly clarified the rules. Absent prescribed parameters, tax lawyers have been left to handicap the chances that a worthlessness deduction will withstand an I.R.S. challenge.

There are several categories, with a declining likelihood of success, of money taxpayers can claim to have lost.

The tax experts consulted for this article universally assigned the highest level of certainty to cash spent to acquire an asset. The roughly $94 million that Mr. Trump’s tax returns show he invested in Chicago fell into this category.

Some gave a lower, though still probable, chance of a taxpayer prevailing in declaring a loss based on loans that a lender agreed to forgive. That’s because forgiven debt generally must be declared as income, which can offset that portion of the worthlessness deduction in the same year. A large portion of Mr. Trump’s worthlessness deduction fell in this category, though he did not begin reporting forgiven debt income until two years later, a delay that would have further reduced his chances of prevailing in an audit.

The tax experts gave the weakest chance of surviving a challenge for a worthlessness deduction based on borrowed money for which the outcome was not clear. It reflects a doubly irrational claim — that the taxpayer deserves a tax benefit for losing someone else’s money even before the money has been lost, and that those anticipated future losses can be used to offset real income from other sources. Most of the debt included in Mr. Trump’s worthlessness deduction was based on that risky position.

Including that debt in the deduction was “just not right,” said Monte Jackel , a veteran of the I.R.S. and major accounting firms who often publishes analyses of partnership tax issues.

A Second Bite at the Apple

Mr. Trump continued to sell units at the Chicago Tower, but still below his costs. Had he done nothing, his 2008 worthlessness deduction would have prevented him from claiming that shortfall as losses again. But in 2010, his lawyers attempted an end-run by merging the entity through which he owned the Chicago tower into another partnership, DJT Holdings L.L.C. In the following years, they piled other businesses, including several of his golf courses, into DJT Holdings.

Those changes had no apparent business purpose. But Mr. Trump’s tax advisers took the position that pooling the Chicago tower’s finances with other businesses entitled him to declare even more tax-reducing losses from his Chicago investment.

His financial problems there continued. More than 100 of the hotel condominiums never sold. Sales of all units totaled only $727 million, far below Mr. Trump’s budgeted costs of $859 million. And some 70,000 square feet of retail space remained vacant because it had been designed without access to foot or vehicle traffic. From 2011 through 2020, Mr. Trump reported $168 million in additional losses from the project.

Those additional write-offs helped Mr. Trump avoid tax liability for his continuing entertainment riches, as well as his unpaid debt from the tower. Starting in 2010, his lenders agreed to forgive about $270 million of those debts. But he was able to delay declaring most of that income until 2014 and spread it out over five years of tax returns, thanks to a provision in the Obama administration’s stimulus bill responding to the Great Recession. In 2018, Mr. Trump reported positive income for the first time in 11 years. But his income tax bill still amounted to only $1.9 million, even as he reported a $25 million gain from the sale of his late father's assets.

It’s unclear when the I.R.S. began to question the 2010 merger transaction, but the conflict escalated during Mr. Trump’s presidency.

The I.R.S. explained its position in a Technical Advice Memorandum , released in 2019, that identified Mr. Trump only as “A.” Such memos, reserved for cases where the law is unclear, are rare and involve extensive review by senior I.R.S. lawyers. The agency produced only two other such memos that year.

The memos are required to be publicly released with the taxpayer’s information removed, and this one was more heavily redacted than usual. Some partnership specialists wrote papers exploring its meaning and importance to other taxpayers, but none identified taxpayer “A” as the then-sitting president of the United States. The Times and ProPublica matched the facts of the memo to information from Mr. Trump’s tax returns and elsewhere.

The 20-page document is dense with footnotes, calculations and references to various statutes, but the core of the I.R.S.’s position is that Mr. Trump’s 2010 merger violated a law meant to prevent double dipping on tax-reducing losses. If done properly, the merger would have accounted for the fact that Mr. Trump had already written off the full cost of the tower’s construction with his worthlessness deduction.

In the I.R.S. memo, Mr. Trump’s lawyers vigorously disagreed with the agency’s conclusions, saying he had followed the law.

If the I.R.S. prevails, Mr. Trump’s tax returns would look very different, especially those from 2011 to 2017. During those years, he reported $184 million in income from “The Apprentice” and agreements to license his name, along with $219 million from canceled debts. But he paid only $643,431 in income taxes thanks to huge losses on his businesses, including the Chicago tower. The revisions sought by the I.R.S. would require amending his tax returns to remove $146 million in losses and add as much as $218 million in income from condominium sales. That shift of up to $364 million could swing those years out of the red and well into positive territory, creating a tax bill that could easily exceed $100 million.

The only public sign of the Chicago audit came in December 2022, when a congressional Joint Committee on Taxation report on I.R.S. efforts to audit Mr. Trump made an unexplained reference to the section of tax law at issue in the Chicago case. It confirmed that the audit was still underway and could affect Mr. Trump’s tax returns from several years.

That the I.R.S. did not initiate an audit of the 2008 worthlessness deduction puzzled the experts in partnership taxation. Many assumed the understaffed I.R.S. simply had not realized what Mr. Trump had done until the deadline to investigate it had passed.

“I think the government recognized that they screwed up,” and then audited the merger transaction to make up for it, Mr. Jackel said.

The agency’s difficulty in keeping up with Mr. Trump’s maneuvers, experts said, showed that this gray area of tax law was too easy to exploit.

“Congress needs to radically change the rules for the worthlessness deduction,” Professor Schwidetzky said.

Susanne Craig contributed reporting.

Read by Eric Jason Martin

Narration produced by Anna Diamond and Krish Seenivasan

Engineered by Steven Szczesniak

Russ Buettner is an investigative reporter. Since 2016, his reporting has focused on the finances of Donald. J. Trump, including articles that revealed tax avoidance schemes evidenced on several decades of his tax returns. In 2019, he shared a Pulitzer Prize for work that revealed the vast inheritance Mr. Trump had received from his father. More about Russ Buettner

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WNBA to expand to Toronto, per report. Team would begin play in 2026.

The WNBA is expanding north of the border.

Toronto has been awarded an WNBA franchise that would begin play in 2026, according to a CBC Sports report that was confirmed by the Toronto Star . It would be owned by Kilmer Sports Inc. and headed by Toronto billionaire Larry Tanenbaum.

Tanenbaum is a minority owner and chairman of Maple Leafs Sports and Entertainment, which owns the NHL Maple Leafs and NBA Raptors, among other teams.

“We continue to engage in productive conversations with interested ownership groups in a number of markets and the granting of any expansion teams requires a vote of the WNBA and NBA Board of Governors,” a WNBA spokesperson said.

A spokesperson for Kilmer Sports said there was "no update at this time."

The expansion team would play at Coca-Cola Coliseum, an 8,000-seat arena that is home to the American Hockey League's Marlies and Toronto's Professional Women's Hockey League franchise.

WNBA commissioner Cathy Engelbert said last April that Toronto was being considered and that she wanted two expansion teams to join the league for play in 2025. The Golden State Warriors were officially awarded a WNBA franchise in October. Toronto, and not Portland, Oregon, appears to be the second choice and 14th franchise. Engelbert said the goal was to reach 16 teams by 2028.

Toronto would become the seventh NBA franchise to have a WNBA team, joining the Brooklyn Nets (New York Liberty), Phoenix Suns (Phoenix Mercury), Minnesota Timberwolves (Minnesota Lynx), Washington Wizards (Washington Mystics, Indiana Pacers (Indiana Fever) and Golden State (team name yet to be announced).