Linear fit

A linear model describes the relationship between a continuous response variable and one or more explanatory variables using a linear function.

• Simple regression models
  Simple regression models describe the relationship between a single predictor variable and a response variable.
• Advanced models
  Advanced models describe the relationship between a response variable and multiple predictor terms.
• Scatter plot
  A scatter plot shows the relationship between variables.
• Summary of fit
  R² and similar statistics measure how much variability is explained by the model.
• Parameter estimates
  Parameter estimates (also called coefficients) are the change in the response associated with a one-unit change of the predictor, all other predictors being held constant.
• Effect of model hypothesis test
  An F-test formally tests the hypothesis of whether the model fits the data better than no model.
• Predicted against actual Y plot
  A predicted against actual plot shows the effect of the model and compares it against the null model.
• Lack of Fit
  An F-test or X²-test formally tests how well the model fits the data.
• Effect of terms hypothesis test
  An F-test formally tests whether a term contributes to the model.
• Effect leverage plot
  An effect leverage plot, also known as added variable plot or partial regression leverage plot, shows the unique effect of a term in the model.
• Effect means
  Effect means are least-squares estimates predicted by the model for each combination of levels in a categorical term, adjusted for the other model effects.
• Multiple comparisons
  Multiple comparisons make simultaneous inferences about a set of parameters.
• Residual plot
  A residual plot shows the difference between the observed response and the fitted response values.
• Residuals - normality
  Normality is the assumption that the underlying residuals are normally distributed, or approximately so.
• Residuals - independence
  Autocorrelation occurs when the residuals are not independent of each other. That is, when the value of e[i+1] is not independent from e[i].
• Outlier and influence plot
  An influence plot shows the outlyingness, leverage, and influence of each case.
• Prediction
  Prediction is the use of the model to predict the population mean or value of an individual future observation, at specific values of the predictors. Inverse prediction deals with the problem of predicting the value of a predictor for a given value of the response variable.

Simple regression models describe the relationship between a single predictor variable and a response variable.

Fit a simple linear regression model to describe the relationship between single a single predictor variable and a response variable.

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click the simple regression model.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the X drop-down list, select the predictor variable.
5. Optional: To assign a weight to each row for the analysis, in the Weight drop-down list, select the weight variable.
6. Click Calculate.

Advanced models describe the relationship between a response variable and multiple predictor terms.

Fit a multiple linear regression model to describe the relationship between many quantitative predictor variables and a response variable.

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click Multiple Regression.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the Available variables list, select the predictor variables:
   • To select a single variable, click the variable.
   • To select multiple variables, click the first variable then hold down the CTRL key and click each additional variable.
   • To select a range of variables, click the first variable then hold down the SHIFT key and click the last variable in the range.
5. Click Add Predictor.
6. Repeat steps 4 through 5 for each additional set of variables.
7. Optional: To assign a weight to each row for the analysis, in the Weight drop-down list, select the weight variable.
8. Click Calculate.

Test if there is a difference between population means when a response variable is classified by one or more categorical variables (factors).

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click ANOVA.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the Available variables list, select the factor variable(s):
   • To select a single variable, click the variable.
   • To select multiple variables, click the first variable then hold down the CTRL key and click each additional variable.
   • To select a range of variables, click the first variable then hold down the SHIFT key and click the last variable in the range.
5. Click Add Factor to add the simple factors.
6. To add the interaction between two or more factors, in the Terms list box, click the first term then hold down the CTRL key and click each additional term to include in the interaction, and then click Cross.
7. To remove a term, in the Terms list box, click the term and then click Remove.
   Note: It is sometimes quicker to use the Factorial command to build a factorial model and then remove any unnecessary terms.
8. To remove a factor and all terms using the factor, in the Factor list box, click the factor and then click Remove.

   When you select a factor, all terms that include the factor are selected. If you hold down the CTRL key and click additional factors, then all terms which include all the selected factors are selected.

9. Repeat steps 4 through 8 to build the model.
10. Click Calculate.

Test if there is a difference between population means when a response variable is classified by two or more categorical variables (factors).

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click ANOVA.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the Available variables list, select the factor variables:
   • To select a single variable, click the variable.
   • To select multiple variables, click the first variable then hold down the CTRL key and click each additional variable.
   • To select a range of variables, click the first variable then hold down the SHIFT key and click the last variable in the range.
5. Click Factorial, and then click Full for a full factorial model, or click the highest order of interaction terms to include in the model.
6. Optional: To sort the terms by order (that is, all simple terms, all two-way interactions, then all three-way interactions...) rather than in the order the variables were added, click Sort.
7. Click Calculate.

Test if there is a difference between population means when a response variable is classified by one or more categorical variables (factors) while adjusting for the effect of one or more quantitative variables (covariates).

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click ANCOVA.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the Available variables list, select the variable(s):
   • To select a single variable, click the variable.
   • To select multiple variables, click the first variable then hold down the CTRL key and click each additional variable.
   • To select a range of variables, click the first variable then hold down the SHIFT key and click the last variable in the range.
5. To add the variables to the model as categorical factors, click Add Factor.
6. To add the variables to the model as quantitative covariates, click Add Covariate.
7. To add the interaction between two or more factors/covariates, in the Terms list box, click the first term then hold down the CTRL key and click each additional term to include in the interaction, and then click Cross.
8. To remove a term, in the Terms list box, click the term and then click Remove.
   Note: It is sometimes quicker to use the Factorial command to build a model and then remove any unnecessary terms.
9. To remove a factor/covariate and all terms using the factor/covariate, in the Factors or Covariates list box, click the factor/covariate and then click Remove.

   When you select a factor or covariate, all terms that include the factor or covariate are selected. If you hold down the CTRL key and click additional factors or covariates, then all terms which include all the selected factors or covariates are selected.

10. Repeat steps 4 through 9 to build the model.
11. Click Calculate.

Fit an advanced linear model to describe the relationship between many predictor variables and a response variable.

1. Select a cell in the dataset.
2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Fit Model, and then click Advanced.
   The analysis task pane opens.
3. In the Y drop-down list, select the response variable.
4. In the Available variables list, select the predictor variable(s):
   • To select a single variable, click the variable.
   • To select multiple variables, click the first variable then hold down the CTRL key and click each additional variable.
   • To select a range of variables, click the first variable then hold down the SHIFT key and click the last variable in the range.
5. Optional: Click the drop-down menu arrow next to the Available variables list, and then select the measurement scale of the variable.

   If the measurement scale is not set, Analyse-it tries to determine if the variable is categorical or continuous scale. After you add a variable to a model, the icon alongside the variable indicates the assumed scale.

6. Click Add to add a variable, or click Factorial to add the factorial terms, or click Polynomial to add a polynomial terms.
7. To add the interaction between two or more variables, in the Terms list box, click the first term then hold down the CTRL key and click each additional term to include in the interaction, and then click Cross.
8. To remove a term, in the Terms list box, click the term and then click Remove.
9. Repeat steps 4 through 8 to build the model.
10. Optional: To assign a weight to each row for the analysis, in the Weight drop-down list, select the weight variable.
11. Click Calculate.

A scatter plot shows the relationship between variables.

The scatter plot identifies the relationship that best describes the data, whether a straight line, polynomial or some other function.

A scatter plot matrix shows the relationship between each predictor and the response, and the relationship between each pair of predictors. You can use the matrix to identify the relationship between variables, to identify where additional terms such as polynomials or interactions are needed, and to see if transformations are needed to make the predictors or response linear. The scatter plot matrix does not convey the joint relationship between each predictor and the response since it does not take into account the effect on the response of the other variables in the model. Effect leverage and residual plots fulfill this purpose after fitting the model.

R² and similar statistics measure how much variability is explained by the model.

R² is the proportion of variability in the response explained by the model. It is 1 when the model fits the data perfectly, though it can only attain this value when all sets of predictors are different. Zero indicates the model fits no better than the mean of the response. You should not use R² when the model does not include a constant term, as the interpretation is undefined.

For models with more than a single term, R² can be deceptive as it increases as you add more parameters to the model, eventually reaching saturation at 1 when the number of parameters equals the number of observations. Adjusted R² is a modification of R² that adjusts for the number of parameters in the model. It only increases when the terms added to the model improve the fit more than would be expected by chance. It is preferred when building and comparing models with a different number of parameters.

For example, if you fit a straight-line model, and then add a quadratic term to the model, the value of R² increases. If you continued to add more the polynomial terms until there are as many parameters as the number of observations, then the R² value would be 1. The adjusted R² statistic is designed to take into account the number of parameters in the model and ensures that adding the new term has some useful purpose rather than simply due to the number of parameters approaching saturation.

In cases where each set of predictor values are not unique, it may be impossible for the R² statistic to reach 1. A statistic called the maximum attainable R² indicates the maximum value that R² can achieve even if the model fitted perfectly. It is related to the pure error discussed in the lack of fit test.

The root mean square error (RMSE) of the fit, is an estimate of the standard deviation of the true unknown random error (it is the square root of the residual mean square). If the model fitted is not the correct model, the estimate is larger than the true random error, as it includes the error due to lack of fit of the model as well as the random errors.

Parameter estimates (also called coefficients) are the change in the response associated with a one-unit change of the predictor, all other predictors being held constant.

The unknown model parameters are estimated using least-squares estimation.

A coefficient describes the size of the contribution of that predictor; a near-zero coefficient indicates that variable has little influence on the response. The sign of the coefficient indicates the direction of the relationship, although the sign can change if more terms are added to the model, so the interpretation is not particularly useful. A confidence interval expresses the uncertainty in the estimate, under the assumption of normally distributed errors. Due to the central limit theorem, violation of the normality assumption is not a problem if the sample size is moderate.

• For quantitative terms, the coefficient represents the rate of change in the response per 1 unit change of the predictor, assuming all other predictors are held constant. The units of measurement for the coefficient are the units of response per unit of the predictor.

  For example, a coefficient for Height of 0.75, in a simple model for the response Weight (kg) with predictor Height (cm), could be expressed as 0.75 kg per cm which indicates a 0.75 kg weight increase per 1 cm in height.

  When a predictor is a logarithm transformation of the original variable, the coefficient is the rate of change in the response per 1 unit change in the log of the predictor. Commonly base 2 log and base 10 log are used as transforms. For base 2 log the coefficient can be interpreted as the rate of change in the response when for a doubling of the predictor value. For base 10 log the coefficient can be interpreted as the rate of change in the response when the predictor is multiplied by 10, or as the % change in the response per % change in the predictor.

• For categorical terms, there is a coefficient for each level:
  • For nominal predictors the coefficients represent the difference between the level mean and the grand mean.

    Analyse-it uses effect coding for nominal terms (also known as the mean deviation coding). The sum of the parameter estimates for a categorical term using effect coding is equal to 0.

  • For ordinal predictors, the coefficients represent the difference between the level mean and the baseline mean.

    Analyse-it uses reference coding for ordinal terms. The first level is used as the baseline or reference level.

• For the constant term, the coefficient is the response when all predictors are 0, and the units of measurement are the same as the response variable.

A standardized parameter estimate (commonly known as standardized beta coefficient) removes the unit of measurement of predictor and response variables. They represent the change in standard deviations of the response for 1 standard deviation change of the predictor. You can use them to compare the relative effects of predictors measured on different scales.

VIF, the variance inflation factor, represents the increase in the variance of the parameter estimate due to correlation (collinearity) between predictors. Collinearity between the predictors can lead to unstable parameter estimates. As a rule of thumb, VIF should be close to the minimum value of 1, indicating no collinearity. When VIF is greater than 5, there is high collinearity between predictors.

A t-test formally tests the null hypothesis that the parameter is equal to 0, against the alternative hypothesis that it is not equal to 0. When the p-value is small, you can reject the null hypothesis and conclude that the parameter is not equal to 0 and it does contribute to the model.

When a parameter is not deemed to contribute statistically to the model, you can consider removing it. However, you should be cautious of removing terms that are known to contribute by some underlying mechanism, regardless of the statistical significance of a hypothesis test, and recognize that removing a term can alter the effect of other terms.

An F-test formally tests the hypothesis of whether the model fits the data better than no model.

It is common to test whether the model fits the data better than a null model that just fits the mean of the response. An analysis of variance table partitions the total variance in the response variable into the variation after fitting the full model (called the model error, or residual), the variation after fitting the null model, and the reduction by fitting the full model compared to the null model.

An F-test formally tests whether the reduction is statistically significant. The null hypothesis states that all the parameters except the intercept are zero against the alternative that at least one parameter is not equal to zero. When the p-value is small, you can reject the null hypothesis and conclude that at least one parameter is not zero.

An analysis of variance (ANOVA) table shows the sources of variation.

A predicted against actual plot shows the effect of the model and compares it against the null model.

For a good fit, the points should be close to the fitted line, with narrow confidence bands. Points on the left or right of the plot, furthest from the mean, have the most leverage and effectively try to pull the fitted line toward the point. Points that are vertically distant from the line represent possible outliers. Both types of points can adversely affect the fit.

The plot is also a visualization of the ANOVA table, except each observation is shown so you can gain much more insight than just the hypothesis test. The confidence interval around the full model line portrays the F-test that all the parameters except the intercept are zero. When the confidence interval does not include the horizontal null model line, the hypothesis test is significant.

An F-test or X²-test formally tests how well the model fits the data.

When the model fitted is correct the residual (model error) mean square provides and unbiased estimate of the true variance. If the model is wrong, then the mean square is larger than the true variance. It is possible to test for lack of fit by comparing the model error mean square to the true variance.

When the true variance is known, a X²-squared test formally tests whether the model error is equal to the hypothesized value.

When the true variance is unknown, and there are multiple observations for each set of predictor values, an F-test formally tests whether there is a difference between pure error and the model error. The pure error is the pooled variance calculated for all unique sets of predictor values. It makes no assumptions about the model but does assume that the variance is the same for each set of predictor values. An analysis of variance table shows the pure error, model error, and the difference between them called the lack of fit.

The null hypothesis states that the model error mean square is equal to the hypothesized value/pure error, against the alternative that it is greater than. When the test p-value is small, you can reject the null hypothesis and conclude that there is a lack of fit.

An F-test formally tests whether a term contributes to the model.

In most modeling analyses the aim is a model that describes the relationship using as few terms as possible. It is therefore of interest to look at each term in the model to decide if the term is providing any useful information.

An analysis of variance table shows the reduction in error by including each term. An F-test for each term is a formal hypothesis test to determine if the term provides useful information to the model. The null hypothesis states that the term does not contribute to the model, against the alternative hypothesis that it does. When the p-value is small, you can reject the null hypothesis and conclude that the term does contribute to the model.

When a term is not deemed to contribute statistically to the model, you may consider removing it. However, you should be cautious of removing terms that are known to contribute by some underlying mechanism, regardless of the statistical significance of a hypothesis test, and recognize that removing a term can alter the effect of other terms.

An effect leverage plot, also known as added variable plot or partial regression leverage plot, shows the unique effect of a term in the model.

A horizontal line shows the constrained model without the term; a slanted line shows the unconstrained model with the term. The plot shows the unique effect of adding a term to a model assuming the model contains all the other terms and the influence of each point on the effect of term hypothesis test. Points further from the horizontal line than the slanted line effectively try to make the hypothesis test more significant, and those closer to the horizontal than the slanted line try to make the hypothesis test less significant. When the confidence band fully encompasses the horizontal line, you can conclude the term does not contribute to the model. It is equivalent to a non-significant F-test for the partial effect of the term. 

You can also use the plot to identify cases that are likely to influence the parameter estimates. Points furthest from the intersection of the horizontal and slanted lines have high leverage, and effectively try to pull the line towards them. A high leverage point that is distant from the bulk of the points can have a large influence the parameter estimates for the term.

Finally, you can use the plot to spot near collinearity between terms. When the terms are collinear, the points collapse toward a vertical line.

Effect means are least-squares estimates predicted by the model for each combination of levels in a categorical term, adjusted for the other model effects.

Least-squares means are predictions made at specific values of the predictors. For each categorical variable in the term, the value is the combination of levels. For all other variables that are not part of the term, the values are neutral values. The neutral value for a quantitative variable is the mean. The neutral value of a nominal variable is the average over all levels which is equivalent to values of 0 for each dummy predictor variable in an effect coding scheme. The neutral value of an ordinal variable is the reference level.

Plot the main effects and interactions to interpret the means.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, in the Terms group, click Effect Means.
   The analysis task pane Effect Means panel opens.
3. To show main effect plots for each factor, click the Main effect plots check box.
4. To show interaction effect plots for each 2-way interaction and up to a single 3-way interaction, click the Interaction effect plots check box.
5. To show the means plot for individual terms, in the Individual effects list box, select the individual term check boxes, and then select the Means plot check box.
6. To show the means table for individual terms, in the Individual effects list box, select the individual term check boxes, and then select the Means check box.
   Note: For convenience, the Means and Means plot options apply to all selected Individual effects.
7. Click Recalculate.

Multiple comparisons make simultaneous inferences about a set of parameters.

When making inferences about more than one parameter (such as comparing many means, or the differences between many means), you must use multiple comparison procedures to make inferences about the parameters of interest. The problem when making multiple comparisons using individual tests such as Student's t-test applied to each comparison is the chance of a type I error increases with the number of comparisons. If you use a 5% significance level with a hypothesis test to decide if two groups are significantly different, there is a 5% probability of observing a significant difference that is simply due to chance (a type I error). If you made 20 such comparisons, the probability that one or more of the comparisons is statistically significant simply due to chance increases to 64%. With 50 comparisons, the chance increases to 92%. Another problem is the dependencies among the parameters of interest also alter the significance level. Therefore, you must use multiple comparison procedures to maintain the simultaneous probability close to the nominal significance level (typically 5%).

Multiple comparison procedures are classified by the strength of inference that can be made and the error rate controlled. A test of homogeneity controls the probability of falsely declaring any pair to be different when in fact all are the same. A stronger level of inference is confident inequalities and confident directions which control the probability of falsely declaring any pair to be different regardless of the values of the others. An even stronger level is a set of simultaneous confidence intervals that guarantees that the simultaneous coverage probability of the intervals is at least 100(1-alpha)% and also have the advantage of quantifying the possible effect size rather than producing just a p-value. A higher strength inference can be used to make an inference of a lesser strength but not vice-versa. Therefore, a confidence interval can be used to perform a confidence directions/inequalities inference, but a test of homogeneity cannot make a confidence direction/inequalities inference.

The most well known multiple comparison procedures, Bonferroni and Šidák, are not multiple comparison procedures per se. Rather they are an inequality useful in producing easy to compute multiple comparison methods of various types. In most scenarios, there are more powerful procedures available. A useful application of Bonferroni inequality is when there are a small number of pre-planned comparisons. In these cases, you can use the standard hypothesis test or confidence interval with the significance level (alpha) set to the Bonferroni inequality (alpha divided by the number of comparisons).

A side effect of maintaining the significance level is a lowering of the power of the test. Different procedures have been developed to maintain the power as high as possible depending on the strength of inference required and the number of comparisons to be made. All contrasts comparisons allow for any possible contrast; all pairs forms the k*(k-1)/2 pairwise contrasts, whereas with best forms k contrasts each with the best of the others, and against control forms k-1 contrasts each against the control group. You should choose the appropriate contrasts of interest before you perform the analysis, if you decide after inspecting the data, then you should only use all contrasts comparison procedures.

Methods of controlling the Type I error and dependencies between point estimates when making multiple comparisons between effect means in a linear model.

Compare the effect means to make inferences about the differences between them.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, in the Terms group, click Effect Means.
   The analysis task pane Effect Means panel opens.
3. In the Individual effects list box, select the individual term check box, and then select the Multiple Comparisons check box.
4. In the Type drop-down list, select the comparisons, and then:
   • If comparing with best, in the Best drop-down list, select the "best" group.
   • If comparing against a control, in the Control drop-down list, select the control group.
5. Optional: To sort the contrasts by the magnitude of the difference, select the Ordered check box.
6. Optional: In the Method drop-down list, select the multiple comparisons procedure, and in the Confidence interval edit box, type the confidence level as a percentage, or type - to suppress the confidence interval, and then in the drop-down list, select the confidence bounds.
7. Optional: To perform a hypothesis test to test if the difference is equal to 0, select the Hypothesis test check box, and then in the Significance level edit box, type the predefined significance level against.
8. Optional: To show a mean-mean scatter plot, select the Mean-Mean scatter plot check box.
   Note: For convenience the confidence interval, hypothesis test, and mean-mean scatter plot options apply to all selected Individual effects.
9. Click Calculate.

A residual plot shows the difference between the observed response and the fitted response values.

The ideal residual plot, called the null residual plot, shows a random scatter of points forming an approximately constant width band around the identity line.

It is important to check the fit of the model and assumptions – constant variance, normality, and independence of the errors, using the residual plot, along with normal, sequence, and lag plot.

For a model with many terms, it can be difficult to identify specific problems using the residual plot. A non-null residual plot indicates that there are problems with the model, but not necessarily what these are.

Normality is the assumption that the underlying residuals are normally distributed, or approximately so.

While a residual plot, or normal plot of the residuals can identify non-normality, you can formally test the hypothesis using the Shapiro-Wilk or similar test.

The null hypothesis states that the residuals are normally distributed, against the alternative hypothesis that they are not normally-distributed. If the test p-value is less than the predefined significance level, you can reject the null hypothesis and conclude the residuals are not from a normal distribution. If the p-value is greater than the predefined significance level, you cannot reject the null hypothesis.

Violation of the normality assumption only becomes an issue with small sample sizes. For large sample sizes, the assumption is less important due to the central limit theorem, and the fact that the F- and t-tests used for hypothesis tests and forming confidence intervals are quite robust to modest departures from normality.

Autocorrelation occurs when the residuals are not independent of each other. That is, when the value of e[i+1] is not independent from e[i].

While a residual plot, or lag-1 plot allows you to visually check for autocorrelation, you can formally test the hypothesis using the Durbin-Watson test. The Durbin-Watson statistic is used to detect the presence of autocorrelation at lag 1 (or higher) in the residuals from a regression. The value of the test statistic lies between 0 and 4, small values indicate successive residuals are positively correlated. If the Durbin-Watson statistic is much less than 2, there is evidence of positive autocorrelation, if much greater than 2 evidence of negative autocorrelation.

Plot the residuals to check the fit and assumptions of the model.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, in the Diagnostics group, click Residuals, and then click:

   Option	Description
   Residual	Plot the residuals.
   3-up	Plot the residuals and a histogram and normal plot.
   5-up	Plot the residuals, a histogram and normal plot, a sequence plot, and a lag plot.

   The analysis task pane, the Residuals panel opens.
3. In the Residuals drop-down list, select:

   Option	Description
   Raw	Plot the raw residual Y - Y fitted.
   Standardized	Plot the standardized residual (Y - Y fitted) / SE.

4. Optional: To test the normality of the residuals, On the Analyse-it ribbon tab, in the Diagnostics group, click Test Normality.
5. Optional: To test for autocorrelation amongst the residuals, On the Analyse-it ribbon tab, in the Diagnostics group, click Test Autocorrelation.
6. Click Recalculate.

An influence plot shows the outlyingness, leverage, and influence of each case.

The plot shows the residual on the vertical axis, leverage on the horizontal axis, and the point size is the square root of Cook's D statistic, a measure of the influence of the point.

Outliers are cases that do not correspond to the model fitted to the bulk of the data. You can identify outliers as those cases with a large residual (usually greater than approximately +/- 2), though not all cases with a large residual are outliers and not all outliers are bad. Some of the most interesting cases may be outliers.

Leverage is the potential for a case to have an influence on the model. You can identify points with high leverage as those furthest to the right. A point with high leverage may not have much influence on the model if it fits the overall model without that case.

Influence combines the leverage and residual of a case to measure how the parameter estimates would change if that case were excluded. Points with a large residual and high leverage have the most influence. They can have an adverse effect on (perturb) the model if they are changed or excluded, making the model less robust. Sometimes a small group of influential points can have an unduly large impact on the fit of the model.

Plot measures to identify cases with large outliers, high leverage, or major influence on the fitted model.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, in the Diagnostics group, click Outliers and Influence.
   The analysis task pane Outliers, Leverage, Influence panel opens.
3. Optional: To highlight and label cases that are large outliers, have high leverage or major influence on the fitted model, select the Identify influential cases check box.
4. Optional: To individually plot the Studentized residual, leverage, and Cook's D influence measures by observation number, select the Index plots check box.
5. Click Recalculate.

Prediction is the use of the model to predict the population mean or value of an individual future observation, at specific values of the predictors. Inverse prediction deals with the problem of predicting the value of a predictor for a given value of the response variable.

When making predictions, it is important that the data used to fit the model is similar to future populations to which you want to apply the prediction. You should be careful of making predictions outside the range of the observed data. Assumptions met for the observed data may not be met outside the range. Non-constant variance can cause confidence intervals for the predicted values to become unrealistically narrow or so large as to be useless. Alternatively, a different fit function may better describe the unobserved data outside the range.

When making multiple predictions of the population mean at different sets of predictor values the confidence intervals can be simultaneous or individual. A simultaneous interval ensures you achieve the confidence level simultaneously for all predictions, whereas individual intervals only ensure confidence for the individual prediction. With individual inferences, the chance of at least one interval not including the true value increases with the number of predictions.

Predict the value of the mean at specific values of the predictors, or the value of an individual future observation.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, click Predict, and then click Y.
   The analysis task pane Predict Y panel opens.
3. In the Terms list box, under the #1 column, type the value for each predictor.

   You can leave a value blank in which case the average value is used. Leaving the value blank is useful in ANOVA models where you can predict the mean at specific combinations averaged over other effects.

4. To predict the expected mean, select the Mean check box, and in the Confidence interval edit box, type the confidence level. Finally, in the Method drop-down list, select Simultaneous for intervals that ensure you achieve the confidence level simultaneously for all predictions, or Individual for intervals only that ensure confidence for each individual prediction.
5. To predict the future value of an individual observation, select the Individual check box, and in the Prediction interval edit box, type the confidence level.
6. Optional: On the Fit panel, in the Predicted values drop-down list, select the style to plot the predicated values on the scatter plot.
7. Click Recalculate.

Predict the values of the predictors at a specific value or values of the response variable.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, click Predict, and then click X given Y.
   The analysis task pane Inverse Predict X given Y panel opens.
3. In the Predictions list box, under the Y column, type the values for the response variable.
4. Optional: In the Confidence interval edit box, type the confidence level, and then in the drop-down list, select the confidence bounds. Finally, in the Method drop-down list, select Simultaneous for intervals that ensure you achieve the confidence level simultaneously for all predictions, or Individual for intervals only that ensure confidence for each individual prediction.
5. Optional: On the Fit panel, in the Predicted values drop-down list, select the style to plot the predicated values on the scatter plot.
6. Click Recalculate.

Save the values computed by the analysis back to the dataset for further analysis.

1. Activate the analysis report worksheet.
2. On the Analyse-it ribbon tab, click Save Variable, and then select the variable to store to the dataset.
   Note: If the Save Variable command is shaded, On the Analyse-it ribbon tab, in the Report group, click Recalculate to update the analysis.

Fit model analysis study requirements and dataset layout.