Data Set: Diving into Data: How to Prepare Your Data Set for Linear Regression in Excel

1. Introduction to Linear Regression and Its Importance in Data Analysis

Linear regression stands as a cornerstone within the world of data analysis, offering a gateway into the realm of predictive analytics. This statistical method enables us to understand and quantify the relationship between two or more variables, typically by fitting a linear equation to observed data. The equation for a simple linear regression model can be expressed as $$ y = \beta_0 + \beta_1x + \epsilon $$, where \( y \) represents the dependent variable we aim to predict, \( x \) denotes the independent variable, \( \beta_0 \) is the y-intercept, \( \beta_1 \) is the slope of the line, and \( \epsilon \) is the error term.

The importance of linear regression in data analysis cannot be overstated. It provides a clear and quantifiable way to measure how changes in one variable are associated with changes in another. This is particularly useful in business and economics, where it can help answer questions like how sales are affected by advertising spend, or in healthcare, where it might be used to understand how a drug dosage impacts recovery time.

From a data scientist's perspective, linear regression is often the first tool in their arsenal when it comes to exploring relationships within data. It's not just the simplicity of its application that makes it a go-to method, but also its interpretability. The coefficients of the regression equation give direct insights into the nature of the relationship between variables.

Let's delve deeper into the nuances of linear regression and its pivotal role in data analysis:

1. Assumptions of linear regression: Before applying linear regression, it's crucial to ensure that certain assumptions are met. These include linearity, independence, homoscedasticity, and normal distribution of residuals. Violating these assumptions can lead to inaccurate models and misleading results.

2. Types of Linear Regression:

- Simple Linear Regression: Involves a single independent variable and is used to establish a linear relationship with the dependent variable.

- multiple Linear regression: Incorporates two or more independent variables, allowing for a more complex analysis of how various factors influence the outcome.

3. Model Evaluation: After fitting a linear regression model, it's essential to evaluate its performance. Metrics like R-squared, adjusted R-squared, and the F-test are commonly used to assess the model's explanatory power and overall fit.

4. Overfitting and Underfitting: These are two common pitfalls in regression analysis. Overfitting occurs when a model is too complex and captures the noise along with the signal. Underfitting happens when the model is too simple and fails to capture the underlying trend.

5. Regularization Techniques: To prevent overfitting, techniques like Ridge Regression (L2 regularization) and Lasso Regression (L1 regularization) can be employed. These methods add a penalty to the regression model, encouraging simpler models that generalize better to new data.

To illustrate the power of linear regression, consider the example of a retail company analyzing the impact of customer satisfaction scores on sales revenue. By collecting data on customer satisfaction and corresponding sales figures, a linear regression model can be constructed to predict how changes in customer satisfaction might influence future sales. This insight can then be used to make informed decisions about where to allocate resources to maximize revenue.

Linear regression is an indispensable tool in the data analyst's toolkit. Its ability to provide actionable insights from data makes it a fundamental technique for anyone looking to make data-driven decisions. Whether you're forecasting sales, evaluating marketing strategies, or exploring scientific phenomena, linear regression offers a clear path to understanding the dynamics at play.

2. What is a Data Set?

At the heart of any data-driven analysis lies the data set, a collection of data points organized for a specific purpose. In the realm of linear regression, particularly when using tools like Excel, understanding the structure and composition of a data set is crucial. It's the foundation upon which predictive models are built, allowing us to extrapolate insights and make informed decisions. A data set can be as simple as a two-column spreadsheet with independent and dependent variables, or as complex as a multi-dimensional array with numerous variables interacting in subtle ways.

From a statistician's perspective, a data set is a sample from a larger population, a snapshot that captures the essential characteristics we're interested in examining. For a computer scientist, it's an array or a matrix, a structured form of data that can be efficiently processed and analyzed by algorithms. Meanwhile, a business analyst might see a data set as a goldmine of insights, trends, and patterns that can drive strategic decisions.

When preparing a data set for linear regression in Excel, consider the following points:

1. Data Cleaning: Before any analysis, ensure your data is clean. This means checking for and handling missing values, outliers, and errors. For example, if you're analyzing sales data, you might find some entries with negative values, which could indicate returns or data entry errors.

2. Variable Selection: Choose which variables to include. In linear regression, you have one dependent variable (the outcome you're trying to predict) and one or more independent variables (the predictors). For instance, if you're trying to predict house prices, your dependent variable could be the price, while independent variables might include square footage, number of bedrooms, and location.

3. Data Formatting: Excel requires data to be in a specific format for regression analysis. Your independent variables should be in separate columns, and your dependent variable should be in the last column. Each row represents a different observation or data point.

4. Data Transformation: Sometimes, transforming data can lead to better models. This could involve normalizing data, creating dummy variables for categorical data, or log-transforming skewed data. For example, if you're working with income data that is heavily skewed, a log transformation can normalize the distribution, making it more suitable for linear regression.

5. Checking Assumptions: linear regression has several key assumptions, such as linearity, independence, homoscedasticity, and normality of residuals. Use Excel's charting and diagnostic tools to check these assumptions. Scatter plots can help assess linearity and independence, while residual plots can help check for homoscedasticity and normality.

6. Running the Regression: Use Excel's Data Analysis Toolpak to run the regression. Input your data range, specify the dependent and independent variables, and choose the output options. Excel will generate a summary output with key statistics like R-squared, coefficients, and p-values.

7. Interpreting Results: Understand what the output means. The R-squared value tells you how much of the variation in the dependent variable is explained by the model. Coefficients indicate the relationship between each independent variable and the dependent variable. P-values help you determine the statistical significance of those relationships.

By considering these aspects, you can prepare your data set for a robust linear regression analysis in excel. Remember, the quality of your input data directly affects the reliability of your output, so meticulous preparation is key to uncovering valuable insights.

3. Setting Up Your Spreadsheet for Success

Embarking on a journey through data analysis in excel, particularly for linear regression, requires a solid foundation. This begins with setting up your spreadsheet efficiently and effectively. A well-organized spreadsheet not only streamlines the process of data analysis but also minimizes errors and simplifies troubleshooting. It's akin to preparing a canvas before painting; the initial effort pays dividends throughout the analytical process. From the perspective of a data analyst, a spreadsheet is a tool for storytelling, where numbers and formulas replace words. For a statistician, it's a laboratory where data sets come to life, revealing trends and patterns. And for a business professional, it's a decision-making engine, driving strategies and insights.

Here are some in-depth steps to ensure your spreadsheet is primed for success:

1. Start with a Clear Structure: Before inputting any data, outline the structure of your spreadsheet. This includes defining which data goes into each column and ensuring that each row represents a unique data point. For example, if you're analyzing sales data, you might have columns for date, product, region, and sales amount.

2. Use Tables: Convert your data range into a table by using the 'Format as Table' feature. This provides powerful tools for sorting and filtering, and it automatically adjusts formulas as you add new data. For instance, if you're tracking monthly expenses, a table will allow you to add new entries without manually updating range references in your formulas.

3. Name Your Ranges: Assign names to specific ranges or cells that you'll use frequently. This makes formulas easier to read and maintain. For example, naming the range containing sales figures as 'TotalSales' allows you to use '=SUM(TotalSales)' instead of '=SUM(B2:B100)'.

4. Implement Data Validation: To maintain data integrity, use data validation rules to restrict the type of data or the values that users can enter into certain cells. If you're expecting a percentage, for instance, set a data validation rule that only allows numbers between 0 and 1.

5. Leverage Conditional Formatting: Highlight important data points using conditional formatting. This can help you quickly identify outliers or trends at a glance. For example, you could highlight all values above a certain threshold in green to indicate high performance.

6. Utilize PivotTables: When it comes to summarizing large data sets, PivotTables are your best friend. They allow you to quickly reorganize and summarize selected columns and rows of data. For a sales dataset, a PivotTable can help you see total sales by product or by month without writing complex formulas.

7. Keep Formulas Simple: While Excel is capable of handling complex formulas, simplicity is key. Break down complicated calculations into smaller, more manageable parts. This not only makes your spreadsheet easier to audit but also reduces the likelihood of errors.

8. Document Your Work: Use comments or a separate 'Documentation' sheet to explain complex formulas or to note the sources of your data. This is crucial for transparency and for those times when you—or someone else—need to revisit the spreadsheet after a long period.

9. Backup Your Data: Regularly save copies of your spreadsheet, especially before making major changes. This ensures that you have a fallback option in case of accidental data loss or corruption.

10. Test Your Setup: Before diving into the analysis, test your spreadsheet with sample data to ensure everything works as expected. This can help catch issues early on, saving you time in the long run.

By following these steps, you'll create a robust and reliable spreadsheet that serves as a strong foundation for your linear regression analysis. Remember, the goal is to make your data work for you, not the other way around. With a well-prepared spreadsheet, you're one step closer to uncovering the insights hidden within your data.

4. Identifying and Handling Outliers and Missing Values

In the realm of data analysis, the integrity and quality of your data set are paramount, especially when preparing for a technique as sensitive to anomalies as linear regression. Before you can explore the relationships between variables and make predictions, you must ensure that your data is clean and reliable. This means rigorously identifying and handling outliers—those data points that fall far outside the expected range—and missing values, which can skew your results and lead to inaccurate models.

Outliers can be the result of errors in data collection, natural variation, or they might indicate something significant that merits further investigation. They can distort statistical analyses, such as the mean or standard deviation, leading to erroneous conclusions. On the other hand, missing values can arise from issues in data extraction, entry, or collection. They pose a challenge because simply ignoring them can bias your results, while different imputation methods can influence your analysis in various ways.

Here's an in-depth look at how to tackle these issues:

1. Outlier Detection:

- Statistical Methods: Use statistical measures like the Z-score or IQR (Interquartile Range) to detect outliers. For instance, a Z-score above 3 or below -3 is often considered an outlier.

- Visualization: box plots and scatter plots can help visualize outliers. Data points that appear far from others can be investigated.

- Domain Knowledge: Sometimes, what appears to be an outlier may be a valid data point. Domain expertise is crucial to discern these cases.

2. Outlier Handling:

- Trimming: Remove outliers from your data set, but be cautious as this can lead to loss of information.

- Transformation: Apply transformations like log or square root to reduce the impact of extreme values.

- Binning: Group data into bins to smooth out minor fluctuations.

3. Missing Value Detection:

- Complete Case Analysis: Listwise deletion of any row with a missing value, suitable when the dataset is large and missing data is minimal.

- Missing Value Indicators: Sometimes, the fact that a value is missing can be informative. Adding a binary indicator can capture this.

4. Missing Value Imputation:

- Mean/Median/Mode Imputation: Replace missing values with the central tendency measure of the variable.

- Prediction Models: Use algorithms like k-Nearest Neighbors or regression models to predict and fill in missing values.

- Multiple Imputation: A more sophisticated approach that involves creating several different imputed datasets and combining the results.

Example: Imagine you're analyzing the relationship between advertising spend and sales. You notice one data point where the spend is \$10 million, but sales are zero. This could be an outlier due to data entry error. After consulting domain experts, you confirm it's an error and decide to remove it from your analysis.

In another case, you have missing sales data for a particular month. Instead of discarding the entire row, you use a predictive model to estimate sales based on the spend and trends from other months.

By carefully identifying and handling outliers and missing values, you ensure that your linear regression model in Excel is built on a solid foundation, leading to more accurate and reliable insights. Remember, data cleaning is not just a preliminary step; it's a critical part of the data analysis process that requires attention to detail and an understanding of the nuances of your data set.

5. Choosing the Right Predictors for Your Model

Variable selection is a critical step in the modeling process because it has a significant impact on the performance of your model. The right set of variables can lead to a model that is both accurate and interpretable, while the wrong set can lead to a model that is either overfitted or underfitted. This means that the model either captures random noise in the data or fails to capture the underlying trends, respectively. The goal of variable selection is to find the best subset of predictors that provides the most reliable representation of the data without overcomplicating the model.

From a statistical perspective, variable selection is about finding the balance between bias and variance. A model with too many predictors may have low bias but high variance, leading to overfitting. Conversely, a model with too few predictors may have high bias and low variance, leading to underfitting. The challenge is to find the "sweet spot" where both bias and variance are minimized.

From a business standpoint, variable selection is about identifying the most meaningful predictors that align with business objectives. For instance, a marketing team might be interested in variables that are directly related to customer behavior and sales, rather than all available data.

Here are some key considerations for variable selection:

1. Relevance: The variable should have a logical reason for being included in the model. It should be related to the outcome you are trying to predict.

- Example: When predicting house prices, square footage is a relevant variable as larger homes typically cost more.

2. Collinearity: Avoid variables that are highly correlated with each other, as they can cause instability in the model coefficients.

- Example: If you have both 'age' and 'age squared' in your model, they are likely to be collinear.

3. Parsimony: The principle of parsimony, or Occam's razor, suggests that among competing models that explain the data equally well, the simplest model is preferred.

- Example: If two models offer similar accuracy, choose the one with fewer variables.

4. Predictive Power: Include variables that contribute significantly to the predictive power of the model.

- Example: A variable that divides customers into segments based on purchasing behavior may have strong predictive power for future sales.

5. Data Quality: Variables should be selected based on the quality of data available. Missing values, outliers, and measurement errors can all affect model performance.

- Example: A variable with 30% missing values might not be reliable unless an effective imputation strategy is in place.

6. Interactions and Non-linearity: Consider the possibility of interactions between variables or non-linear relationships.

- Example: The effect of marketing spend on sales might increase at a decreasing rate, suggesting a non-linear relationship.

7. External Validity: The selected variables should make sense not just within the dataset but also in the real world.

- Example: economic indicators like gdp growth rate might be relevant for predicting market trends.

8. Computational Efficiency: More variables mean more computation. If the model is to be used in a real-time application, computational efficiency could be a constraint.

- Example: A real-time bidding system in online advertising requires a fast and efficient model.

9. Regulatory and Ethical Considerations: Ensure that the variables used do not infringe on privacy laws or ethical standards.

- Example: Avoid using sensitive personal information like race or religion as predictors.

In summary, variable selection is not just a statistical exercise; it is a multifaceted process that involves statistical, business, and ethical considerations. By carefully selecting the right predictors, you can build a robust and effective model that serves its purpose well. Remember, the goal is not just to fit the data you have but to predict new data as accurately as possible.

6. Ensuring Consistency for Accurate Analysis

In the realm of data analysis, the process of formatting data is a critical step that cannot be overstated. It is the meticulous task of transforming raw data into a structured format that is consistent and easily interpretable by both humans and software. This step is particularly crucial when preparing a data set for linear regression analysis in Excel, as it ensures that the data inputs are accurate, which in turn, guarantees the reliability of the output. Linear regression is sensitive to inconsistencies; even a single outlier or incorrectly formatted entry can skew the results, leading to erroneous conclusions. Therefore, formatting must be approached with a keen eye for detail and a deep understanding of the data at hand.

From the perspective of a data scientist, formatting involves standardizing units, aligning decimal points, and ensuring that categorical data is properly encoded. For instance, if the data set includes a variable for temperature, it's essential to decide whether to use Celsius or Fahrenheit and apply that choice uniformly across the entire data set. Similarly, for a variable like 'State of Residence,' one must decide between using full state names or abbreviations and then apply that consistently.

From the standpoint of a business analyst, formatting data is about making the data set comprehensible and ready for strategic decision-making. It involves organizing data in a way that highlights trends and patterns pertinent to business goals. For example, sales data might be formatted to show monthly totals for each product category, facilitating a clear comparison over time.

Here are some key steps to ensure data is formatted correctly for linear regression in excel:

1. Uniformity in Data Types: Ensure that all data in a column is of the same type. Numbers should not be mixed with text, and dates should be in a consistent format.

2. Categorical Variable Encoding: Convert categorical variables into a numerical format using methods like one-hot encoding, where each category is represented by a binary column.

3. Handling Missing Values: Decide on a strategy for missing data, whether it's removing those rows, replacing with the mean/median, or using a more complex imputation method.

4. Scaling and Normalization: If the range of values varies widely between variables, consider scaling (changing the range of data) or normalization (changing the distribution of data).

5. Outlier Detection and Treatment: Identify and address outliers that can affect the regression model's accuracy. Options include removal, capping, or transformation.

6. data validation: Use Excel's data Validation feature to restrict data entry to certain types or ranges, which can prevent formatting errors.

7. Use of Conditional Formatting: Highlight important data points, like potential outliers or errors, using Excel's Conditional Formatting tool.

For example, consider a data set with the variables 'Age' and 'Income' for a population sample. If 'Age' is recorded in years and 'Income' in thousands of dollars, it's important to format these consistently. If 'Income' has some entries in dollars and others in thousands, this inconsistency can lead to incorrect analysis. By ensuring that all 'Income' data is in the same unit, the linear regression model can accurately interpret the relationship between 'Age' and 'Income'.

Formatting data for linear regression in Excel is a foundational step that paves the way for accurate analysis. It requires a blend of technical skills and contextual understanding to ensure that the data set is a true reflection of the phenomena being studied. By adhering to these principles, one can confidently move forward in the data analysis process, knowing that the foundation is solid.

7. Training and Testing Sets for Model Validation

In the realm of data analysis, particularly when employing linear regression models, the practice of splitting your dataset into training and testing sets is a critical step that cannot be overstated. This process is akin to a teacher preparing a student for an examination; the training set is used to impart knowledge and skills, while the testing set evaluates the student's ability to apply what they've learned to new, unseen problems. The training set serves as the foundation upon which your model learns the underlying patterns within your data, allowing it to make predictions. Conversely, the testing set acts as an unbiased evaluator of your model's predictive performance on data it has never encountered before. This dichotomy is essential for model validation, ensuring that your model is not only memorizing the data (a phenomenon known as overfitting) but is genuinely learning from it and can generalize its insights to new data.

From a statistical perspective, the division of data into training and testing sets allows us to mitigate the risk of overfitting. Overfitting occurs when a model performs exceptionally well on the training data but fails to predict future observations accurately. By holding out a portion of the data as a testing set, we can assess the model's performance on data that was not used during the training phase, providing a more accurate measure of its predictive power.

Here are some in-depth insights into the process:

1. Proportion of Split: The typical split ratio ranges from 70/30 to 80/20 between training and testing sets, respectively. However, this can vary based on the size of your dataset. For smaller datasets, a 10-fold cross-validation might be more appropriate, where the data is split into ten parts, and the model is trained and tested across all possible combinations.

2. Random Sampling: It's crucial to randomly sample the data when creating training and testing sets to avoid any bias. Tools like Excel's RAND() function or programming languages like Python's train_test_split from sklearn.model_selection can be used for this purpose.

3. Stratification: When dealing with datasets that have a significant imbalance in the output variable, stratified sampling ensures that both training and testing sets have a proportional representation of each class. This is vital for maintaining the integrity of the model's evaluation.

4. Time Series Split: For time-dependent data, a simple random split is not suitable. Instead, a time series split must be used where the training set consists of past data and the testing set comprises future data, respecting the chronological order.

5. Feature Scaling: Before splitting, it's important to perform feature scaling to ensure that all variables contribute equally to the model's performance. This can be done using standardization or normalization techniques.

6. Dimensionality Reduction: Techniques like principal Component analysis (PCA) can be applied before the split to reduce the number of features, especially if the dataset has a large number of variables, which can help in improving model performance and reducing overfitting.

To illustrate these points, let's consider an example using Excel for a dataset with housing prices. After applying the RAND() function to shuffle the data, we might allocate the first 80% of the rows to the training set and the remaining 20% to the testing set. We would then use the training set to build our linear regression model, perhaps using Excel's data Analysis toolpak. After training, we would apply the model to the testing set and compare the predicted housing prices against the actual prices to evaluate the model's accuracy.

By adhering to these principles, we ensure that our linear regression model in Excel is robust, reliable, and ready to provide valuable insights from our data.

8. Step-by-Step Guide in Excel

Linear regression is a foundational statistical technique that allows us to model the relationship between a dependent variable and one or more independent variables. It's particularly useful in the realm of data analysis for making predictions or understanding the underlying trends within a data set. Excel, with its robust set of analytical tools, provides a convenient platform for implementing linear regression, even for those who may not have a strong background in statistics.

1. Data Preparation: Before diving into the regression analysis, ensure your data is clean and organized. Remove any outliers or erroneous data points that could skew your results. Structure your data in a tabular format with independent variables in separate columns and the dependent variable in a single column.

2. Plotting Data: visualize the relationship between the variables by creating scatter plots. This can give you a preliminary idea of the correlation and help in identifying any patterns or anomalies.

3. Inserting the Regression Tool: Navigate to the 'Data' tab and click on 'Data Analysis.' If you don't see this option, you'll need to enable the 'Analysis ToolPak' add-in from Excel options. Select 'Regression' from the list of analysis tools.

4. Configuring the Regression Tool: In the regression dialog box, specify the 'Input Y Range' as your dependent variable and the 'Input X Range' as your independent variable(s). If you have multiple independent variables, ensure they are contiguous columns.

5. Interpreting the Output: Excel will generate an output table that includes several statistical measures. The key metrics to look at are the R-squared value, which indicates the proportion of variance explained by the model, and the p-values for each coefficient, which test the significance of each independent variable.

6. Analyzing Residuals: The residuals, or differences between the observed and predicted values, should be randomly distributed. Use Excel's residual plots to check for any patterns that might suggest a poor fit.

7. Refining the Model: Based on the analysis, you may need to adjust your model. This could involve transforming variables, adding interaction terms, or removing insignificant predictors.

8. Prediction: Once satisfied with the model, use the regression equation to make predictions. Excel can calculate these predictions automatically using the 'FORECAST.LINEAR' function.

For example, let's say we have a data set that includes the number of hours studied (independent variable) and the exam scores (dependent variable) for a group of students. After preparing the data and plotting it, we notice a positive trend, suggesting that more hours studied leads to higher exam scores. We then run the regression analysis in excel and find that our R-squared value is 0.85, indicating a strong model. The p-value for the hours studied is less than 0.05, confirming its significance in predicting exam scores. With this model, we can now predict the exam score for a student who studies a certain number of hours, enhancing our understanding of the study habits and performance relationship.

By following these steps and utilizing Excel's capabilities, you can effectively implement linear regression to uncover valuable insights from your data, guiding informed decision-making and strategy development.

9. Making Sense of Your Linear Regression Output

Once you've run your linear regression analysis in Excel, the output can seem daunting with its array of numbers, statistics, and charts. However, this output is a goldmine of insights, waiting to be interpreted and transformed into actionable business intelligence. The key to unlocking these insights lies in understanding what each part of the output is telling you about the relationship between your independent (predictor) variables and the dependent (outcome) variable.

1. Coefficients: At the heart of the output are the coefficients. These numbers represent the change in the dependent variable for a one-unit change in the predictor. For example, if you're predicting house prices based on square footage, a coefficient of 3000 for square footage would mean that for each additional square foot, the house price increases by $3000.

2. R-squared value: This statistic tells you how much of the variance in your dependent variable is explained by the model. An R-squared value of 0.70 means that 70% of the variability in your outcome can be explained by the predictors. It's a measure of the model's goodness-of-fit.

3. P-values: Associated with each coefficient, the p-value tells you whether the relationships observed in your data are statistically significant. A common threshold for significance is 0.05. If a predictor's p-value is below this, you can be confident that there is a statistically significant association between this predictor and the outcome variable.

4. Confidence Intervals: These intervals give you a range within which you can be confident the true coefficient lies, with a certain level of confidence, usually 95%. If a confidence interval for a coefficient does not include zero, it suggests that there is a significant relationship between the predictor and the outcome.

5. Residuals: The difference between the observed values and the values predicted by the model are the residuals. Analyzing the pattern of residuals can tell you a lot about the appropriateness of your model. For instance, if residuals are randomly scattered around zero, it suggests your model is a good fit for the data.

6. F-statistic: This is a measure of the overall significance of the model. It compares the model with no predictors (only an intercept) to your model to determine if your predictors are collectively significant.

7. Standard Error: This statistic gives you an idea of the average distance that the observed values fall from the regression line. A smaller standard error suggests a more precise estimate of the coefficient.

8. T-statistics: Similar to the p-value, the t-statistic helps determine if a coefficient is significantly different from zero. It's used to calculate the p-value.

9. durbin-Watson statistic: This number helps detect the presence of autocorrelation in the residuals from a regression analysis. Values close to 2 suggest no autocorrelation, while values deviating far from 2 indicate it.

10. Multicollinearity: It's important to check for multicollinearity, which occurs when predictor variables are correlated with each other. This can be assessed through variance Inflation factor (VIF) values, where a VIF above 10 indicates high multicollinearity.

By carefully examining each of these components, you can interpret your linear regression output to make informed decisions. Remember, the goal is not just to have a statistically significant model, but one that is also meaningful and useful for your specific context. Always consider the practical significance of your findings alongside the statistical significance. For instance, a statistically significant predictor with a very small effect size might not be of practical importance in a business setting. Conversely, a predictor with a large effect size but just above the significance threshold might still be worth considering for its potential impact.

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