P value: Beyond a Shadow of a Doubt: The Significance of P values in Linear Regression

1. Introduction to P-values and Their Role in Hypothesis Testing

P-values are a cornerstone of statistical hypothesis testing, providing a measure to help determine the strength of the results. They are used to test the validity of a null hypothesis, which is a default statement that there is no effect or no difference. When we perform a linear regression analysis, we're often interested in understanding whether the relationships we observe in our data are statistically significant or if they could have occurred by random chance. This is where p-values come into play, offering a bridge between the data and the inferences we draw from it.

From a frequentist perspective, a p-value is the probability of observing a test statistic as extreme as, or more extreme than, the value observed, assuming that the null hypothesis is true. If this probability is very low, we have evidence against the null hypothesis and may consider it to be statistically significant. In the context of linear regression, a low p-value for a coefficient suggests that the corresponding predictor is significantly associated with the response variable.

However, p-values are often misunderstood and misused. They are not the probability that the null hypothesis is true, nor are they the probability that the alternative hypothesis is true. They do not tell us the size of an effect or its practical significance. They simply give us a tool for deciding whether the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Let's delve deeper into the role of p-values in hypothesis testing with a focus on linear regression:

1. Null Hypothesis (H0): In the context of linear regression, the null hypothesis typically states that there is no relationship between the independent variable and the dependent variable. For example, ( H_0: \beta_1 = 0 ) suggests that the slope coefficient ( \beta_1 ) is zero, indicating no linear relationship.

2. alternative hypothesis (H1): The alternative hypothesis is what you want to prove. For instance, ( H_1: \beta_1 \neq 0 ) posits that there is a non-zero association between the variables.

3. Test Statistic: The test statistic is calculated from the data and is used to decide whether to reject the null hypothesis. In linear regression, the t-statistic for a coefficient is computed as ( t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} ), where ( \hat{\beta}_1 ) is the estimated coefficient and ( SE(\hat{\beta}_1) ) is its standard error.

4. P-value: The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis.

5. Interpretation: If the p-value is less than or equal to the significance level (α), we reject the null hypothesis. For example, if \( p \leq 0.05 \), we would say there is statistically significant evidence to suggest a relationship between the variables.

6. Examples: Consider a study examining the effect of study hours on exam scores. The linear regression might yield a p-value of 0.03 for the slope coefficient of study hours. This would suggest that there is only a 3% chance of observing such a strong association between study hours and exam scores if there were no actual relationship, providing evidence to reject the null hypothesis.

P-values are a vital part of hypothesis testing in linear regression, helping us to make decisions about the significance of our findings. However, they should be interpreted with caution and in conjunction with other metrics such as confidence intervals and effect sizes to understand the full picture of the data's story.

2. A Primer

Linear regression is a foundational statistical method that allows us to quantify and interpret the relationships between two or more variables. At its core, linear regression models the linear relationship between a dependent variable and one or more independent variables. This model assumes that the dependent variable can be explained by a linear combination of the independent variables, plus an error term that captures random fluctuations or noise in the data.

From a statistical perspective, linear regression is used to make predictions, understand relationships, and identify trends. It's a tool that's as useful in business for forecasting sales as it is in science for analyzing experimental data. The beauty of linear regression lies in its simplicity and interpretability. The coefficients of the independent variables represent the expected change in the dependent variable for a one-unit change in the independent variable, holding all other variables constant.

However, the simplicity of linear regression also brings with it certain assumptions and limitations. For instance, it assumes that there is a linear relationship between the variables, that the residuals are normally distributed and homoscedastic, and that the independent variables are not too highly correlated with each other—a condition known as multicollinearity.

Let's delve deeper into the nuances of linear regression:

1. The Equation: The linear regression model can be represented by the equation $$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n + \epsilon $$ where \( y \) is the dependent variable, \( \beta_0 \) is the intercept, \( \beta_1, \beta_2, ..., \beta_n \) are the coefficients, \( x_1, x_2, ..., x_n \) are the independent variables, and \( \epsilon \) is the error term.

2. Interpreting Coefficients: Each coefficient \( \beta_i \) represents the average effect on \( y \) of a one-unit increase in \( x_i \), assuming all other variables are held constant. For example, in a model predicting house prices, a coefficient of 3000 for square footage would suggest that for each additional square foot, we expect the house price to increase by $3000, all else being equal.

3. The Role of P-values: In the context of linear regression, p-values help us determine the statistical significance of each coefficient. A low p-value (typically less than 0.05) indicates that we can reject the null hypothesis that the coefficient is zero (i.e., it has no effect).

4. Assumptions: For the model to be valid, several assumptions must be met, including linearity, independence, homoscedasticity, and normal distribution of residuals. If these assumptions are violated, the model's predictions and inferences may not be reliable.

5. Diagnostics and Residual Analysis: After fitting a linear regression model, it's crucial to perform diagnostic checks. Residual plots can reveal patterns that suggest violations of assumptions, such as non-linearity or heteroscedasticity.

6. Multicollinearity: When independent variables are highly correlated, it can be difficult to determine the individual effect of each variable on the dependent variable. This is known as multicollinearity, and it can inflate the variance of the coefficient estimates, making them unstable and unreliable.

7. Model Selection: Choosing the right variables for your model is both an art and a science. Techniques like forward selection, backward elimination, and stepwise regression can help in identifying the most significant variables.

8. Regularization: Methods like ridge Regression and lasso Regression introduce a penalty term to the regression equation to prevent overfitting and to handle multicollinearity by shrinking the coefficients of less important variables.

To illustrate these concepts, consider a simple example where we're trying to predict the fuel efficiency of cars based on their engine size and weight. Our linear regression model might look something like this:

$$ \text{Fuel Efficiency} = \beta_0 + \beta_1 \times \text{Engine Size} + \beta_2 \times \text{Weight} + \epsilon $$

If we find that the p-value associated with the coefficient for engine size is less than 0.05, we might conclude that there is a statistically significant relationship between engine size and fuel efficiency. However, if the weight variable has a high p-value, we might decide it's not a significant predictor in the presence of engine size and consider removing it from the model.

In summary, linear regression is a powerful and versatile tool, but it requires careful application and interpretation. Understanding the underlying assumptions, being mindful of potential pitfalls, and using diagnostic tools to validate the model are all critical steps in decoding the linear regression model.

3. Understanding Regression Coefficients

In the realm of linear regression, the dance of variables is a delicate interplay where each step, each movement, represents the intricate relationship between independent variables and the dependent variable they predict. Regression coefficients are the choreographers of this dance, dictating the direction and intensity of the relationship. These coefficients, often symbolized as beta values in statistical models, tell a story of how a single unit change in an independent variable is expected to affect the dependent variable, holding all other variables constant.

This dance is not a solo performance; it's a group number where each variable has a role, and understanding these roles is crucial for interpreting the results of a regression analysis. From the perspective of a statistician, a regression coefficient is a measure of association, a numerical value that represents the strength and direction of the relationship. Economists might view these coefficients as elasticities, reflecting the responsiveness of one economic variable to changes in another. In the field of psychology, these values can indicate the magnitude of an effect, such as how much an intervention might improve a clinical outcome.

Let's delve deeper into the nuances of regression coefficients:

1. Standardized vs. Unstandardized Coefficients: Standardized coefficients, also known as beta weights, are the coefficients obtained when the data have been standardized. This means they have been scaled to have a mean of zero and a standard deviation of one. These coefficients are useful for comparing the relative importance of variables within a model. In contrast, unstandardized coefficients are raw coefficients calculated from the original data. They provide an estimate of the actual change in the dependent variable for a one-unit change in the independent variable.

2. Interpreting Coefficients with Different Scales: When variables are on different scales, interpreting coefficients can be challenging. For example, consider a regression model predicting house prices with the size of the house (in square feet) and the number of bedrooms. A coefficient for size might be much smaller than the coefficient for the number of bedrooms simply because the unit of measurement for size is much smaller. To compare the impact of these variables, one might look at standardized coefficients or rescale the variables to be on similar scales.

3. Interaction Effects: Sometimes, the effect of one independent variable on the dependent variable depends on the level of another independent variable. This is where interaction terms come into play. For instance, the effect of education level on income might be different for men and women. Including an interaction term between gender and education level in the regression model can help uncover these nuanced relationships.

4. Multicollinearity: When independent variables are highly correlated with each other, it can cause multicollinearity, which inflates the variance of the regression coefficients. This makes them unstable and difficult to interpret. Detecting multicollinearity often involves looking at correlation matrices or variance inflation factors (VIFs).

5. Coefficient Significance: The significance of a regression coefficient is typically assessed using a t-test, which tests the null hypothesis that the coefficient is equal to zero. A significant coefficient suggests that there is evidence to believe that the variable has an effect on the dependent variable. However, significance does not imply causality; it merely indicates an association.

To illustrate these concepts, let's consider a hypothetical example. Imagine a study examining the factors that influence the success of startup companies. The regression model might include variables such as initial capital, the experience of the founding team, and the competitiveness of the market. The coefficient for initial capital might be positive, indicating that, on average, startups with more initial capital tend to be more successful. However, if there's an interaction effect between initial capital and market competitiveness, the model might reveal that the positive effect of capital is stronger in less competitive markets.

Understanding regression coefficients is akin to learning the steps of a complex dance. Each step is meaningful, and only by considering the choreography as a whole can one appreciate the beauty and intricacy of the performance. By examining regression coefficients through various lenses and considering the context of the variables involved, we can gain a richer understanding of the data and the stories they tell.

4. The Gatekeeper of Statistical Significance

In the realm of statistical analysis, particularly within the framework of linear regression, the p-value stands as a sentinel, guarding the threshold of statistical significance. This seemingly innocuous probability measure is the linchpin in determining whether the observed relationships in data are genuine or merely the result of random fluctuations. It's a concept that has garnered both reverence and skepticism, as it can both illuminate and mislead, depending on its interpretation and context.

From the perspective of a traditional statistician, the p-value is a sacred benchmark. It quantifies the probability of observing a result at least as extreme as the one obtained, assuming that the null hypothesis is true. In essence, it's a gauge of compatibility between the observed data and the null hypothesis. A low p-value, typically less than 0.05, suggests that the observed data is unlikely under the null hypothesis, prompting analysts to reject the null in favor of the alternative hypothesis.

However, critics argue that the p-value is often misused and misinterpreted. They point out that a p-value does not measure the probability that the null hypothesis is true, nor does it indicate the size or importance of an effect. Moreover, the arbitrary threshold of 0.05 has been a subject of contention, with some advocating for a more stringent cutoff to reduce the rate of false discoveries, while others propose abandoning the p-value altogether in favor of other metrics like effect size or confidence intervals.

To delve deeper into the intricacies of p-values in linear regression, consider the following points:

1. Definition and Calculation: The p-value is calculated from the test statistic, which in linear regression is often the t-statistic. This statistic measures how many standard deviations the estimated coefficient is away from zero. The p-value is then derived from the t-distribution, reflecting the probability of obtaining a t-statistic as extreme as the observed one under the null hypothesis.

2. Interpretation: A common misstep is to interpret the p-value as the probability that the null hypothesis is true. Instead, it should be viewed as a measure of evidence against the null hypothesis. A smaller p-value indicates stronger evidence to reject the null.

3. Contextual Relevance: The significance of a p-value must be considered in the context of the study. For instance, in fields where the consequences of a false positive are high, such as in medical trials, a more conservative p-value threshold might be warranted.

4. Multiple Comparisons: In studies involving multiple tests, the chance of finding a statistically significant result by chance increases. Techniques like the Bonferroni correction adjust p-values to account for multiple comparisons, reducing the likelihood of type I errors.

5. Effect Size and Confidence Intervals: While the p-value can indicate whether an effect exists, it does not convey the magnitude of the effect. Supplementing p-values with confidence intervals and measures of effect size provides a more comprehensive understanding of the results.

To illustrate these points, let's consider an example from a hypothetical linear regression analysis. Suppose a researcher is investigating the relationship between hours of study and exam scores among students. The regression output yields a p-value of 0.03 for the coefficient representing hours of study. This p-value suggests that there is only a 3% chance of observing such a strong association between study hours and exam scores if, in reality, no such association exists. However, without considering the effect size, which might reveal that each additional hour of study only increases the exam score by a negligible amount, the practical significance of this finding remains unclear.

While the p-value is a crucial tool in the arsenal of statistical analysis, it is imperative to wield it with care and discernment. By considering it alongside other statistical measures and within the broader context of the research, we can navigate the nuanced landscape of statistical significance with greater precision and reliability.

5. What Does Significant Really Mean?

In the realm of statistical analysis, particularly within linear regression, the concept of significance is often encapsulated by the p-value. This metric is a pivotal indicator, guiding researchers in determining whether their findings can be considered statistically significant or merely the result of random chance. However, the interpretation of p-values is nuanced and can be subject to misinterpretation. It's not merely a passport to significance but rather a complex measure that requires careful consideration.

From a traditional standpoint, a p-value that falls below a predetermined threshold, commonly 0.05, suggests that the observed effect is statistically significant, implying that the likelihood of the effect occurring by chance alone is less than 5%. Yet, this binary interpretation has been critiqued for its oversimplification of statistical evidence. Critics argue that it neglects the magnitude and practical significance of the effect, as well as the context of the study.

Here are some insights from different perspectives:

1. The Frequentist Perspective: This view considers the p-value as the probability of observing data as extreme as the sample given that the null hypothesis is true. It's a measure of how incompatible the data are with the null hypothesis, not the probability that the hypothesis is correct.

2. The Bayesian Approach: Bayesians interpret the p-value in light of prior beliefs and evidence. They argue for the integration of prior knowledge with the observed data, which can lead to different conclusions about significance.

3. The Practical Significance: Beyond statistical significance, researchers must consider the real-world relevance of their results. An effect might be statistically significant but too small to be of any practical use.

4. The Reproducibility Concern: A significant p-value does not guarantee that the results can be replicated in future studies. The reproducibility crisis in science has led to calls for more stringent thresholds or alternative metrics.

5. The Misuse of P-values: P-values can be misused, leading to 'p-hacking' where researchers manipulate their study to achieve a low p-value. This undermines the integrity of statistical conclusions.

To illustrate these points, let's consider an example. Suppose a study finds that a new drug reduces blood pressure by an average of 2 mmHg with a p-value of 0.04. From a Frequentist perspective, this result is significant. However, a Bayesian might incorporate prior studies showing no effect and conclude the significance is overstated. From a practical standpoint, a 2 mmHg reduction might not be clinically meaningful. The reproducibility concern would prompt skepticism until the results are replicated. Lastly, if the study was designed with multiple interim analyses, the risk of p-hacking must be considered.

Interpreting p-values demands a multifaceted approach that transcends the traditional threshold-based method. It requires a balance between statistical rigor and practical relevance, always within the context of the broader research landscape.

6. Common Misconceptions About P-values in Regression Analysis

P-values are a staple of statistical significance in regression analysis, yet they are often misunderstood and misinterpreted. This section delves into the common misconceptions surrounding p-values, aiming to clarify their true meaning and proper use in the context of linear regression. P-values are not a definitive measure of 'truth' or 'falsehood' but rather an indicator of how incompatible the data are with a specified statistical model. They do not measure the probability that the studied hypothesis is true, nor do they convey the size of an effect or the importance of a result. Understanding p-values requires a nuanced appreciation of statistical theory and a critical approach to data analysis.

Misconception 1: A low p-value confirms the research hypothesis.

A low p-value indicates that the observed data would be unlikely under the null hypothesis. It does not prove the research hypothesis; it merely suggests that the data are not consistent with the null hypothesis.

Example: If a study finds a p-value of 0.03 when testing the effect of a new drug, it means that there is a 3% chance of observing such data if the drug had no effect. It does not mean there is a 97% chance that the drug is effective.

Misconception 2: The p-value can tell us the probability that the results occurred by chance.

The p-value measures how well the data fit with the null hypothesis, not the chance occurrence of the results. It is a conditional probability, not an absolute measure of certainty.

Example: A p-value of 0.05 does not imply a 5% probability that the results are due to random chance; rather, it means that there is a 5% probability of obtaining at least as extreme results given that the null hypothesis is true.

Misconception 3: P-values provide a 'yes-or-no' answer to hypothesis tests.

P-values do not offer a binary conclusion but rather inform the degree of evidence against the null hypothesis. They should be interpreted in the context of the research question and other study information.

Example: A p-value of 0.049 is not substantially different from a p-value of 0.051, though the former might be considered 'significant' and the latter 'not significant' in many research contexts.

Misconception 4: A non-significant p-value means the null hypothesis is true.

A non-significant p-value indicates that the data do not provide strong evidence against the null hypothesis, but it does not confirm the null hypothesis is true.

Example: If a study on a diet's effect on weight loss yields a p-value of 0.2, it suggests that the data do not provide strong evidence against the null hypothesis of no effect. However, it does not prove that the diet has no effect on weight loss.

Misconception 5: P-values and statistical significance are the end-all of research findings.

Statistical significance does not equate to practical significance. Researchers must consider the effect size, confidence intervals, and the real-world implications of the findings.

Example: A study might find a statistically significant effect of a new teaching method on student performance (p < 0.05), but if the effect size is small, the practical implications of adopting the new method may be limited.

P-values are a complex statistical concept that require careful interpretation. They are not standalone measures but part of a larger analytical framework that includes effect sizes, confidence intervals, and theoretical considerations. By dispelling these misconceptions, researchers can better communicate the significance of their findings and avoid overreliance on p-values as a measure of research validity.

7. Complementary Measures to Assess Model Fit

In the realm of statistical analysis, particularly within the context of linear regression, the P-value has long been the cornerstone for determining the significance of results. However, the reliance on P-values alone has been a subject of intense debate. Critics argue that P-values can be misleading, failing to convey the true reliability of the model, especially when used in isolation. This has led to a growing consensus among statisticians and researchers for the adoption of complementary measures to assess model fit.

One such measure is the Adjusted R-squared, which modifies the R-squared value to account for the number of predictors in the model. Unlike the regular R-squared, which can artificially inflate with the addition of more variables, the Adjusted R-squared provides a more honest reflection of the model's explanatory power.

Information Criteria like the akaike Information criterion (AIC) and the bayesian Information criterion (BIC) offer another perspective. These criteria penalize model complexity, thus favoring models that achieve a good fit with fewer parameters. The lower the AIC or BIC value, the better the model is considered to be at balancing fit and simplicity.

Residual analysis is also pivotal. By examining the residuals – the differences between observed and predicted values – one can check for non-random patterns that suggest poor model fit. For instance, a plot of residuals should ideally show no discernible pattern; if a pattern exists, it indicates that the model is missing a key variable or that there is a non-linear relationship.

Let's delve deeper into these measures:

1. Adjusted R-squared: It adjusts the statistic based on the number of predictors in the model, helping to prevent overfitting. For example, a model with an R-squared of 0.8 might seem impressive, but if it includes hundreds of predictors, the Adjusted R-squared will be significantly lower, signaling potential overfitting.

2. AIC and BIC: Both criteria help in model selection. A practical example is comparing two models where Model A has an AIC of 300 and Model B has an AIC of 280. Despite Model A having a slightly higher R-squared, Model B is preferred due to its lower AIC, indicating a better model fit when considering the number of predictors.

3. Residual Plots: These are essential for visualizing the fit of the model. If the residuals are randomly scattered around the horizontal axis, the model is appropriate. However, if the residuals display a pattern, such as a curve, it suggests that the model is not capturing some aspect of the data's structure.

4. Cross-Validation: This technique involves dividing the data into subsets, using some for training and others for validation. The model's performance across these subsets provides a robust assessment of its predictive power. For instance, a model might perform well on the training set but poorly on the validation set, indicating overfitting.

5. Cohen's f^2: This effect size measure complements the P-value by quantifying the magnitude of the relationship. For example, an f^2 value of 0.15 indicates a medium effect size, suggesting that the independent variable has a substantial impact on the dependent variable, regardless of the P-value.

While P-values serve an important role in hypothesis testing, they are not infallible indicators of model fit. By incorporating these complementary measures, researchers can gain a more nuanced understanding of their models, leading to more reliable and insightful conclusions. The key is to use a combination of these tools to triangulate on the truth, rather than relying on any single measure. This multifaceted approach ensures a more robust and credible assessment of model performance, ultimately enriching the scientific discourse.

8. P-values in Action Across Different Industries

P-values have become a cornerstone in the realm of statistical analysis, serving as a bridge between the theoretical world of mathematics and the practical application of research across various industries. They offer a quantifiable measure to assess the strength of the evidence against a null hypothesis. By examining case studies from different sectors, we can appreciate the versatility and profound impact of p-values in validating findings and guiding decision-making processes.

1. Healthcare: In clinical trials, p-values are pivotal in determining the efficacy of new treatments. For instance, a study comparing the recovery rates of two patient groups treated with different drugs might yield a p-value of 0.03. This suggests that there's only a 3% probability that the observed difference in recovery rates is due to chance, indicating a statistically significant improvement with the new drug.

2. Manufacturing: Quality control often relies on p-values to confirm the consistency of product batches. A low p-value in a test comparing the tensile strength of materials from different production runs could signal a significant variation, prompting a review of the manufacturing process to identify and rectify inconsistencies.

3. Agriculture: P-values assist in assessing the effectiveness of fertilizers or genetically modified seeds. An agricultural study might reveal a p-value of 0.05 when comparing crop yields, suggesting that the new seed variant has a statistically significant impact on production, assuming a typical alpha level of 0.05.

4. Finance: In econometrics, p-values help in validating the predictive power of financial models. A regression analysis on stock returns with a p-value below 0.01 for a key predictor variable, such as interest rate changes, would indicate a strong relationship, influencing investment strategies.

5. Marketing: P-values are used to evaluate the success of marketing campaigns. A/B testing often employs p-values to determine if the difference in response rates between two campaign versions is statistically significant, thereby guiding future marketing efforts.

Through these examples, it's evident that p-values are more than just a statistical tool; they are a critical component in the decision-making process across diverse fields, providing a standardized approach to measure the reliability of experimental results. Their application in linear regression and other statistical methods continues to shed light on patterns and relationships that might otherwise remain obscured, reinforcing their role as a fundamental element in the pursuit of empirical knowledge.

9. The Future of P-values in Statistical Analysis

As we reflect on the role of p-values in statistical analysis, particularly in the context of linear regression, it's clear that their future is poised at a crossroads. The p-value has long been a staple of statistical significance testing, offering a quantifiable measure to determine whether the results of an analysis could be due to chance. However, the reliance on this single metric has come under scrutiny. Critics argue that p-values are often misinterpreted, leading to false positives and the replication crisis in science. Proponents, on the other hand, maintain that when used correctly, p-values are an invaluable tool for hypothesis testing.

The debate is not just academic; it has practical implications for how research is conducted and interpreted. As we move forward, the statistical community is exploring new methodologies and refining existing ones to ensure that p-values continue to serve their intended purpose without being the sole arbiter of scientific validity.

1. Re-evaluation of Significance Thresholds: Traditionally, a p-value of 0.05 has been considered the threshold for statistical significance. However, some researchers suggest that this is an arbitrary cutoff and advocate for a more stringent threshold, such as 0.005, to reduce the rate of false discoveries.

2. Bayesian Approaches: There's a growing interest in Bayesian statistics, which provide a different framework for interpreting data. Unlike p-values, Bayesian methods allow for the incorporation of prior knowledge and can offer a more nuanced understanding of results.

3. Effect Size and Confidence Intervals: Alongside p-values, there's a push to emphasize the reporting of effect sizes and confidence intervals. These metrics provide context to the magnitude and precision of an effect, which p-values alone do not convey.

4. Transparency and Reproducibility: The future of p-values is also tied to the broader movement towards open science. By sharing data and methodologies, researchers can verify results, which helps to address the issue of p-value fishing and p-hacking.

5. Education and Training: Proper interpretation of p-values requires a solid understanding of statistics. Enhancing education around statistical concepts can help prevent common pitfalls in the use of p-values.

For example, consider a study examining the effect of a new drug on blood pressure. A p-value of 0.04 might lead one to conclude that the drug is effective. However, without considering the effect size, we don't know if the drug's impact is clinically significant. If the effect size is small and the confidence interval wide, the practical importance of the finding is questionable.

The future of p-values in statistical analysis is not about discarding them but rather about integrating them into a more holistic approach to data interpretation. By combining p-values with other statistical measures, fostering a culture of transparency, and improving statistical literacy, we can enhance the reliability and credibility of scientific research. The journey ahead for p-values is one of evolution, not obsolescence.

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