# Interpret all statistics and graphs for Simple Regression

Find definitions and interpretation guidance for every statistic and graph that is provided with the simple regression analysis.

## 95% CI

The confidence interval for the fit provides a range of likely values for the mean response given the specified settings of the predictors.

### Interpretation

Use the confidence interval to assess the estimate of the fitted value for the observed values of the variables.

For example, with a 95% confidence level, you can be 95% confident that the confidence interval contains the population mean for the specified values of the variables in the model. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. A wide confidence interval indicates that you can be less confident about the mean of future values. If the interval is too wide to be useful, consider increasing your sample size.

## 95% PI

The prediction interval is a range that is likely to contain a single future response for a value of the predictor variable.

### Interpretation

With 95% prediction bands, you can be 95% confident that new observations will fall within the interval indicated by the purple lines. (Note, however, that this is only true for density values that are within the range included in the analysis.)

For example, a materials engineer at a furniture manufacturing site develops a simple regression model to predict the stiffness of particleboard from the density of the board. The engineer verifies that the model meets the assumptions of the analysis. Then, the analyst uses the model to predict the stiffness.

The regression equation predicts that the stiffness for a new observation with a density of 25 is -21.53 + 3.541*25, or 66.995. While it is unlikely that such an observation would have a stiffness of exactly 66.995, the prediction interval indicates that the engineer can be 95% confident that the actual value will be between approximately 48 and 86.

The prediction interval is always wider than the corresponding confidence interval. In this example, the 95% confidence interval indicates that the engineer can be 95% confident that the mean stiffness will be between approximately 60 and 74.

Adjusted mean squares measure how much variation a term or a model explains, assuming that all other terms are in the model, regardless of the order they were entered. Unlike the adjusted sums of squares, the adjusted mean squares consider the degrees of freedom.

The adjusted mean square of the error (also called MSE or s2) is the variance around the fitted values.

### Interpretation

Minitab uses the adjusted mean squares to calculate the p-value for a term. Minitab also uses the adjusted mean squares to calculate the adjusted R2 statistic. Usually, you interpret the p-values and the adjusted R2 statistic instead of the adjusted mean squares.

Adjusted sums of squares are measures of variation for different components of the model. The order of the predictors in the model does not affect the calculation of the adjusted sum of squares. In the Analysis of Variance table, Minitab separates the sums of squares into different components that describe the variation due to different sources.

The regression sum of squares is the sum of the squared deviations of the fitted response values from the mean response value. It quantifies the amount of variation in the response data that is explained by the model.
The adjusted sum of squares for a term is the increase in the regression sum of squares compared to a model with only the other terms. It quantifies the amount of variation in the response data that is explained by each term in the model.
The error sum of squares is the sum of the squared residuals. It quantifies the variation in the data that the predictors do not explain.
The total sum of squares is the sum of the regression sum of squares and the error sum of squares. It quantifies the total variation in the data.

### Interpretation

Minitab uses the adjusted sums of squares to calculate the p-value for a term. Minitab also uses the sums of squares to calculate the R2 statistic. Usually, you interpret the p-values and the R2 statistic instead of the sums of squares.

## Coef

A regression coefficient describes the size and direction of the relationship between a predictor and the response variable. Coefficients are the numbers by which the values of the term are multiplied in a regression equation.

### Interpretation

The coefficient of the term represents the change in the mean response for a one-unit change in that term. The sign of the coefficient indicates the direction of the relationship between the term and the response. If the coefficient is negative, as the term increases, the mean value of the response decreases. If the coefficient is positive, as the term increases, the mean value of the response increases.

For example, a manager determines that an employee's score on a job skills test can be predicted using the regression model y = 130 + 4.3x. In the equation, x is the hours of in-house training (from 0 to 20) and y is the test score. The coefficient, or slope, is 4.3, which indicates that, for every hour of training, the test score increases, on average, by 4.3 points.

The size of the coefficient is usually a good way to evaluate the practical significance of the effect that a term has on the response variable. However, the size of the coefficient does not indicate whether a term is statistically significant because the calculations for significance also consider the variation in the response data. To determine statistical significance, examine the p-value for the term.

## DF

The total degrees of freedom (DF) are the amount of information in your data. The analysis uses that information to estimate the values of unknown population parameters. The total DF is determined by the number of observations in your sample. The DF for a term show how much information that term uses. Increasing your sample size provides more information about the population, which increases the total DF. Increasing the number of terms in your model uses more information, which decreases the DF available to estimate the variability of the parameter estimates.

If two conditions are met, then Minitab partitions the DF for error. The first condition is that there must be terms you can fit with the data that are not included in the current model. For example, if you have a continuous predictor with 3 or more distinct values, you can estimate a quadratic term for that predictor. If the model does not include the quadratic term, then a term that the data can fit is not included in the model and this condition is met.

The second condition is that the data contain replicates. Replicates are observations where each predictor has the same value. For example, if you have 3 observations where pressure is 5 and temperature is 25, then those 3 observations are replicates.

If the two conditions are met, then the two parts of the DF for error are lack-of-fit and pure error. The DF for lack-of-fit allow a test of whether the model form is adequate. The lack-of-fit test uses the degrees of freedom for lack-of-fit. The more DF for pure error, the greater the power of the lack-of-fit test.

## Fit

Fitted values are also called fits or . The fitted values are point estimates of the mean response for given values of the predictors. The values of the predictors are also called x-values.

### Interpretation

Fitted values are calculated by entering the specific x-values for each observation in the data set into the model equation.

For example, if the equation is y = 5 + 10x, the fitted value for the x-value, 2, is 25 (25 = 5 + 10(2)).

Observations with fitted values that are very different from the observed value may be unusual. Observations with unusual predictor values may be influential. If Minitab determines that your data include unusual or influential values, your output includes the table of Fits and Diagnostics for Unusual Observations, which identifies these observations. The unusual observations that Minitab labels do not follow the proposed regression equation well. However, it is expected that you will have some unusual observations. For example, based on the criteria for large standardized residuals, you would expect roughly 5% of your observations to be flagged as having a large standardized residual. For more information on unusual values, go to Unusual observations.

## Fitted line plot

The fitted line plot displays the response and predictor data. The plot includes the regression line, which represents the regression equation. You can also choose to display the 95% confidence and prediction intervals on the plot.

### Interpretation

Evaluate how well the model fits your data and whether the model meets your goals. Examine the fitted line plot to determine whether the following criteria are met:
• The sample contains an adequate number of observations throughout the entire range of all the predictor values.
• The model properly fits any curvature in the data. If you fit a linear model and see curvature in the data, repeat the analysis and select the quadratic or cubic model. To determine which model is best, examine the plot and the goodness-of-fit statistics. Check the p-value for the terms in the model to make sure they are statistically significant, and apply process knowledge to evaluate practical significance.
• Look for any outliers, which can have a strong effect on the results. Try to identify the cause of any outliers. Correct any data entry or measurement errors. Consider removing data values that are associated with abnormal, one-time events (special causes). Then, repeat the analysis. For more information on detecting outliers, go to Unusual observations.

## F-value

An F-value appears for each term in the Analysis of Variance table:
F-value for the model or the terms
The F-value is the test statistic used to determine whether the term is associated with the response.
F-value for the lack-of-fit test
The F-value is the test statistic used to determine whether the model is missing higher-order terms that include the predictors in the current model.

### Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

A sufficiently large F-value indicates that the term or model is significant.

If you want to use the F-value to determine whether to reject the null hypothesis, compare the F-value to your critical value. You can calculate the critical value in Minitab or find the critical value from an F-distribution table in most statistics books. For more information on using Minitab to calculate the critical value, go to Using the inverse cumulative distribution function (ICDF) and click "Use the ICDF to calculate critical values".

## Histogram of residuals

The histogram of the residuals shows the distribution of the residuals for all observations.

### Interpretation

Use the histogram of the residuals to determine whether the data are skewed or include outliers. The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
A long tail in one direction Skewness
A bar that is far away from the other bars An outlier

Because the appearance of a histogram depends on the number of intervals used to group the data, don't use a histogram to assess the normality of the residuals. Instead, use a normal probability plot.

A histogram is most effective when you have approximately 20 or more data points. If the sample is too small, then each bar on the histogram does not contain enough data points to reliably show skewness or outliers.

## Normal probability plot of the residuals

The normal plot of the residuals displays the residuals versus their expected values when the distribution is normal.

### Interpretation

Use the normal probability plot of residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.

The following patterns violate the assumption that the residuals are normally distributed.

If you see a nonnormal pattern, use the other residual plots to check for other problems with the model, such as missing terms or a time order effect. If the residuals do not follow a normal distribution, prediction intervals can be inaccurate. If the residuals do not follow a normal distribution and the data have fewer than 15 observations, then confidence intervals for predictions, confidence intervals for coefficients, and p-values for coefficients can be inaccurate.

## P-value – Lack-of-fit

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

### Interpretation

To determine whether the model correctly specifies the relationship between the response and the predictors, compare the p-value for the lack-of-fit test to your significance level to assess the null hypothesis. The null hypothesis for the lack-of-fit test is that the model correctly specifies the relationship between the response and the predictors. Usually, a significance level (denoted as alpha or α) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the model correctly specifies the relationship between the response and the predictors when the model does not.
P-value ≤ α: The lack-of-fit is statistically significant
If the p-value is less than or equal to the significance level, you conclude that the model does not correctly specify the relationship. To improve the model, you may need to add terms or transform your data.
P-value > α: The lack-of-fit is not statistically significant

If the p-value is larger than the significance level, the test does not detect any lack-of-fit.

## P-value – Regression

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

### Interpretation

To determine whether the model explains the variation in the response, compare the p-value for the model to your significance level to assess the null hypothesis. The null hypothesis for the overall regression is that the model does not explain any of the variation in the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the model explains variation in the response when the model does not.
P-value ≤ α: The model explains variation in the response
If the p-value is less than or equal to the significance level, you conclude that the model explains variation in the response.
P-value > α: There is not enough evidence to conclude that the model explains variation in the response

If the p-value is greater than the significance level, you cannot conclude that the model explains variation in the response. You may want to fit a new model.

## P-value – Term

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

### Interpretation

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that the term's coefficient is equal to zero, which indicates that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.
P-value ≤ α: The association is statistically significant
If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
P-value > α: The association is not statistically significant

If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. If you fit a quadratic model or a cubic model and the quadratic or cubic terms are not statistically significant, you may want to select a different model.

## R-sq

R2 is the percentage of variation in the response that is explained by the model. It is calculated as 1 minus the ratio of the error sum of squares (which is the variation that is not explained by model) to the total sum of squares (which is the total variation in the model).

### Interpretation

Use R2 to determine how well the model fits your data. The higher the R2 value, the better the model fits your data. R2 is always between 0% and 100%.

You can use a fitted line plot to graphically illustrate different R2 values. The first plot illustrates a simple regression model that explains 85.5% of the variation in the response. The second plot illustrates a model that explains 22.6% of the variation in the response. The more variation that is explained by the model, the closer the data points fall to the fitted regression line. Theoretically, if a model could explain 100% of the variation, the fitted values would always equal the observed values and all of the data points would fall on the fitted line.
Consider the following issues when interpreting the R2 value:
• R2 always increases when you add additional predictors to a model. For example, the best five-predictor model will always have an R2 that is at least as high the best four-predictor model. Therefore, R2 is most useful when you compare models of the same size.

• Small samples do not provide a precise estimate of the strength of the relationship between the response and predictors. If you need R2 to be more precise, you should use a larger sample (typically, 40 or more).

• R2 is just one measure of how well the model fits the data. Even when a model has a high R2, you should check the residual plots to verify that the model meets the model assumptions.

Adjusted R2 is the percentage of the variation in the response that is explained by the model, adjusted for the number of predictors in the model relative to the number of observations. Adjusted R2 is calculated as 1 minus the ratio of the mean square error (MSE) to the mean square total (MS Total).

### Interpretation

Use adjusted R2 when you want to compare models that have different numbers of predictors. R2 always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R2 value incorporates the number of predictors in the model to help you choose the correct model.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled potato chips per container. You receive the following results as you add the predictors in a forward stepwise approach:
Step % Potato Cooling rate Cooking temp R2 Adjusted R2 P-value
1 X     52% 51% 0.000
2 X X   63% 62% 0.000
3 X X X 65% 62% 0.000

The first step yields a statistically significant regression model. The second step adds cooling rate to the model. Adjusted R2 increases, which indicates that cooling rate improves the model. The third step, which adds cooking temperature to the model, increases the R2 but not the adjusted R2. These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.

## R-sq (pred)

Predicted R2 is calculated with a formula that is equivalent to systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. The value of predicted R2 ranges between 0% and 100%. (While the calculations for predicted R2 can produce negative values, Minitab displays zero for these cases.)

### Interpretation

Use predicted R2 to determine how well your model predicts the response for new observations. Models that have larger predicted R2 values have better predictive ability.

A predicted R2 that is substantially less than R2 may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population, although they may appear important in the sample data. The model becomes tailored to the sample data and therefore, may not be useful for making predictions about the population.

Predicted R2 can also be more useful than adjusted R2 for comparing models because it is calculated with observations that are not included in the model calculation.

For example, an analyst at a financial consulting company develops a model to predict future market conditions. The model looks promising because it has an R2 of 87%. However, the predicted R2 is only to 52%, which indicates that the model may be over-fit.

## Regression equation

Use the regression equation to describe the relationship between the response and the terms in the model. The regression equation is an algebraic representation of the regression line. The regression equation for the linear model takes the following form: y = b0 + b1x1. In the regression equation, y is the response variable, b0 is the constant or intercept, b1 is the estimated coefficient for the linear term (also known as the slope of the line), and x1 is the value of the term.

The regression equation with more than one term takes the following form:

y = b0 + b1x1 + b2x2 + ... + bkxk

In the regression equation, the letters represent the following:
• y is the response variable
• b0 is the constant
• b1, b2, ..., bk are the coefficients
• x1, x2, ..., xk are the values of the term

## Resid

A residual (ei) is the difference between an observed value (y) and the corresponding fitted value, (), which is the value predicted by the model.

### Interpretation

Plot the residuals to determine whether your model is adequate and meets the assumptions of regression. Examining the residuals can provide useful information about how well the model fits the data. In general, the residuals should be randomly distributed with no obvious patterns and no unusual values. If Minitab determines that your data include unusual observations, it identifies those observations in the Fits and Diagnostics for Unusual Observations table in the output. The observations that Minitab labels as unusual do not follow the proposed regression equation well. However, it is expected that you will have some unusual observations. For example, based on the criteria for large residuals, you would expect roughly 5% of your observations to be flagged as having a large residual. For more information on unusual values, go to Unusual observations.

## Residuals versus fits

The residuals versus fits graph plots the residuals on the y-axis and the fitted values on the x-axis.

### Interpretation

Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
Fanning or uneven spreading of residuals across fitted values Nonconstant variance
Curvilinear A missing higher-order term
A point that is far away from zero An outlier
A point that is far away from the other points in the x-direction An influential point
The following graphs show an outlier and a violation of the assumption that the residuals are constant.

If you identify any patterns or outliers in your residual versus fits plot, consider the following solutions:

Issue Possible solution
Nonconstant variance Transform the response variable. You can transform the variable in Minitab Statistical Software.
An outlier or influential point
1. Verify that the observation is not a measurement error or data-entry error.
2. Consider performing the analysis without this observation to determine how it impacts your results.
A missing higher-order term Add the term and refit the model.

## Residuals versus order

The residual versus order plot displays the residuals in the order that the data were collected.

### Interpretation

Use the residuals versus order plot to verify the assumption that the residuals are independent from one another. Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent. Ideally, the residuals on the plot should fall randomly around the center line:
If you see a pattern, investigate the cause. The following types of patterns may indicate that the residuals are dependent.

## Residuals versus variables

The residuals versus variables plot displays the residuals versus another variable. The variable could already be included in your model. Or, the variable may not be in the model, but you suspect it affects the response.

### Interpretation

If the variable is already included in the model, use the plot to determine whether you should add a higher-order term of the variable. If the variable is not already included in the model, use the plot to determine whether the variable is affecting the response in a systematic way.

These patterns can identify an important variable or term.
Pattern What the pattern may indicate
Pattern in residuals The variable affects the response in a systematic way. If the variable is not in your model, include a term for that variable and refit the model.
Curvature in the points A higher-order term of the variable should be included in the model. For example, a curved pattern indicates that you should add a squared term.

## S

S represents how far the data values fall from the fitted values. S is measured in the units of the response.

### Interpretation

Use S to assess how well the model describes the response. S is measured in the units of the response variable and represents the how far the data values fall from the fitted values. The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled potato chips per container. You reduce the model to the significant predictors, and S is calculated as 1.79. This result indicates that the standard deviation of the data points around the fitted values is 1.79. If you are comparing models, values that are lower than 1.79 indicate a better fit, and higher values indicate a worse fit.

## SE Coef

The standard error of the coefficient estimates the variability between coefficient estimates that you would obtain if you took samples from the same population again and again.

### Interpretation

Use the standard error of the coefficient to measure the precision of the estimate of the coefficient. The smaller the standard error, the more precise the estimate. Dividing the coefficient by its standard error calculates a t-value. If the p-value associated with this t-statistic is less than your significance level (denoted as alpha or α), you conclude that the coefficient is statistically significant.

For example, technicians estimate a model for insolation as part of a solar thermal energy test:
Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 809 377 2.14 0.042 South 20.81 8.65 2.41 0.024 2.24 North -23.7 17.4 -1.36 0.186 2.17 Time of Day -30.2 10.8 -2.79 0.010 3.86

In this model, North and South measure the position of a focal point in inches. The coefficients for North and South are similar in magnitude. The standard error of the South coefficient is smaller than that of North. Therefore, the model is able to estimate the coefficient for South with greater precision.

The standard error of the North coefficient is nearly as large as the value of the coefficient itself. The resulting p-value is greater than common levels of the significance level, so you cannot conclude that the coefficient for North differs from 0.

While the coefficient for South is closer to 0 than the coefficient for North, the standard error of the coefficient for South is also smaller. The resulting p-value is smaller than common significance levels. Because the estimate of the coefficient for South is more precise, you can conclude that the coefficient for South differs from 0.

Statistical significance is one criterion you can use to reduce a model in multiple regression. For more information, go to Model reduction.

## Std Resid

The standardized residual equals the value of a residual (ei) divided by an estimate of its standard deviation.

### Interpretation

Use the standardized residuals to help you detect outliers. Standardized residuals greater than 2 and less than −2 are usually considered large. The Fits and Diagnostics for Unusual Observations table identifies these observations with an 'R'. The observations that Minitab labels do not follow the proposed regression equation well. However, it is expected that you will have some unusual observations. For example, based on the criteria for large standardized residuals, you would expect roughly 5% of your observations to be flagged as having a large standardized residual. For more information, go to Unusual observations.

Standardized residuals are useful because raw residuals might not be good indicators of outliers. The variance of each raw residual can differ by the x-values associated with it. This unequal variation causes it to be difficult to assess the magnitudes of the raw residuals. Standardizing the residuals solves this problem by converting the different variances to a common scale.

## T-value

The t-value measures the ratio between the coefficient and its standard error.

### Interpretation

Minitab uses the t-value to calculate the p-value, which you use to test whether the coefficient is significantly different from 0.

You can use the t-value to determine whether to reject the null hypothesis. However, the p-value is used more often because the threshold for the rejection of the null hypothesis does not depend on the degrees of freedom. For more information on using the t-value, go to Using the t-value to determine whether to reject the null hypothesis.

## VIF

The variance inflation factor (VIF) indicates how much the variance of a coefficient is inflated due to the correlations among the predictors in the model.

### Interpretation

Use the VIF to describe how much multicollinearity (which is correlation between predictors) exists in a regression analysis. Multicollinearity is problematic because it can increase the variance of the regression coefficients, making it difficult to evaluate the individual impact that each of the correlated predictors has on the response.

Use the following guidelines to interpret the VIF:
VIF Status of predictors
VIF = 1 Not correlated
1 < VIF < 5 Moderately correlated
VIF > 5 Highly correlated
VIF values greater than 5 suggest that the regression coefficients are poorly estimated due to severe multicollinearity.

For more information on multicollinearity and how to mitigate the effects of multicollinearity, see Multicollinearity in regression.

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