# Assumptions of linear regression regarding residuals

The assumptions for multiple linear regression are largely the same as those for simple linear regression modelsso we recommend that you revise them on Page 2.

However there are a few new issues to think about and it is worth reiterating our assumptions for using multiple explanatory variables. Linear relationship: The model is a roughly linear one.

### Assumptions Of Linear Regression Algorithm

This is slightly different from simple linear regression as we have multiple explanatory variables. This time we want the outcome variable to have a roughly linear relationship with each of the explanatory variablestaking into account the other explanatory variables in the model. Homoscedasticity: Ahhh, homoscedasticity - that word again just rolls off the tongue doesn't it! This can be tested for each separate explanatory variable, though it is more common just to check that the variance of the residuals is constant at all levels of the predicted outcome from the full model i.

Independent errors: This means that residuals should be uncorrelated.

## What are the four assumptions of linear regression?

As with simple regression, the assumptions are the most important issues to consider but there are also other potential problems you should look out for:.

Variance in all predictors: It is important that your explanatory variables Explanatory variables may be continuousordinal or nominal but each must have at least a small range of values even if there are only two categorical possibilities. Multicollinearity: Multicollinearity exists when two or more of the explanatory variables are highly correlated. This is a problem as it can be hard to disentangle which of them best explains any shared variance with the outcome. It also suggests that the two variables may actually represent the same underlying factor. Normally distributed residuals: The residuals should be normally distributed.

You can review the simple linear regression assumptions on Page 2. It is important that you check that each scatterplot is exhibiting a linear relationship between variables perhaps adding a regression line to help you with this.

Alternatively you can just check the scatterplot of the actual outcome variable against the predicted outcome. Now that you're a bit more comfortable with regression and the term residual you may want to consider the difference between outliers and influential cases a bit further.

Have a look at the two scatterplots below Figures 3. Note how the two problematic data points influence the regression line in differing ways.

The simple outlier influences the line to a far lesser degree but will have a very large residual distance to the regression line. SPSS can help you spot outliers by identifying cases with particularly large residuals. The influential case outlier dramatically alters the regression line but might be harder to spot as the residual is small - smaller than most of the other more representative points in fact!

A case this extreme is very rare! As well as examining the scatterplot you can also use influence statistics such as the Cook's distance statistic to identify points that may unduly influence the model.

We will talk about these statistics and how to interpret them during our example. Variance in all explanatory variables: This one is fairly easy to check - just create a histogram for each variable to ensure that there is a range of values or that data is spread between multiple categories. This assumption is rarely violated if you have created good measures of the variables you are interested in. Multicollinearity: The simplest way to ascertain whether or not your explanatory variables are highly correlated with each other is to examine a correlation matrix.

If correlations are above.Topics: Regression Analysis. Regression analysis can be a very powerful tool, which is why it is used in a wide variety of fields. The analysis captures everything from understanding the strength of plastic to the relationship between the salaries of employees and their gender. I've even used it for fantasy football! But there are assumptions your data must meet in order for the results to be valid.

In this article, I'm going to focus on the assumptions that the error terms or "residuals" have a mean of zero and constant variance. When you run a regression analysis, the variance of the error terms must be constant, and they must have a mean of zero.

If this isn't the case, your model may not be valid. To check these assumptions, you should use a residuals versus fitted values plot. Below is the plot from the regression analysis I did for the fantasy football article mentioned above. The errors have constant variance, with the residuals scattered randomly around zero. If, for example, the residuals increase or decrease with the fitted values in a pattern, the errors may not have constant variance. The points on the plot above appear to be randomly scattered around zero, so assuming that the error terms have a mean of zero is reasonable.

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The vertical width of the scatter doesn't appear to increase or decrease across the fitted values, so we can assume that the variance in the error terms is constant.

But what if this wasn't the case? What if we did notice a pattern in the plot? I created some fake data to illustrate this point, then created another plot. There is definitely a noticeable pattern here! The residuals error terms take on positive values with small or large fitted values, and negative values in the middle.

The width of the scatter seems consistent, but the points are not randomly scattered around the zero line from left to right. This graph tells us we should not use the regression model that produced these results. So what to do? There's no single answer, but there are several options. One approach is to adjust your model: adding a squared term to the model could solve the issue with the residuals plot. Alternatively, Minitab has a tool that can adjust the data so that the model is appropriate and will yield acceptable residual plots.

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It's called a Box-Cox transformation, and it's easy to use!We make a few assumptions when we use linear regression to model the relationship between a response and a predictor. These assumptions are essentially conditions that should be met before we draw inferences regarding the model estimates or before we use a model to make prediction.

Because we are fitting a linear model, we assume that the relationship really is linear, and that the errors, or residuals, are simply random fluctuations around the true line.

This is the assumption of equal variance. We also assume that the observations are independent of one another. Correlation between sequential observations, or auto-correlationcan be an issue with time series data -- that is, with data with a natural time-ordering. How do we check regression assumptions?

We examine the variability left over after we fit the regression line. We simply graph the residuals and look for any unusual patterns. This is a graph of each residual value plotted against the corresponding predicted value. If the assumptions are met, the residuals will be randomly scattered around the center line of zero, with no obvious pattern. The residuals will look like an unstructured cloud of points, centered at zero.

If there is a non-random pattern, the nature of the pattern can pinpoint potential issues with the model. For example, if curvature is present in the residuals, then it is likely that there is curvature in the relationship between the response and the predictor that is not explained by our model. A linear model does not adequately describe the relationship between the predictor and the response. In this example, the linear model systematically over-predicts some values the residuals are negativeand under-predict others the residuals are positive.

This means that the variability in the response is changing as the predicted value increases. This is a problem, in part, because the observations with larger errors will have more pull or influence on the fitted model. An unusual pattern might also be caused by an outlier. Outliers can have a big influence on the fit of the regression line. In this example, we have one obvious outlier.

Many of the residuals with lower predicted values are positive these are above the center line of zerowhereas many of the residuals for higher predicted values are negative. The one extreme outlier is essentially tilting the regression line. As a result, the model will not predict well for many of the observations. In addition to the residual versus predicted plot, there are other residual plots we can use to check regression assumptions.

A histogram of residuals and a normal probability plot of residuals can be used to evaluate whether our residuals are approximately normally distributed. Note that we check the residuals for normality. Our response and predictor variables do not need to be normally distributed in order to fit a linear regression model. If the data are time series data, collected sequentially over time, a plot of the residuals over time can be used to determine whether the independence assumption has been met.Stay up-to-date.

Building a linear regression model is only half of the work. In order to actually be usable in practice, the model should conform to the assumptions of linear regression. Though, the X2 is raised to power 2, the equation is still linear in beta parameters. So the assumption is satisfied in this case. Check the mean of the residuals.

If it zero or very closethen this assumption is held true for that model. This is default unless you explicitly make amends, such as setting the intercept term to zero. This produces four plots. The top-left and bottom-left plots shows how the residuals vary as the fitted values increase. From the first plot top-leftas the fitted values along x increase, the residuals decrease and then increase. This pattern is indicated by the red line, which should be approximately flat if the disturbances are homoscedastic.

The plot on the bottom left also checks this, and is more convenient as the disturbance term in Y axis is standardized. In this case, there is a definite pattern noticed. So, there is heteroscedasticity. Lets check this on a different model. Now, the points appear random and the line looks pretty flat, with no increasing or decreasing trend. So, the condition of homoscedasticity can be accepted. This is applicable especially for time series data.

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Autocorrelation is the correlation of a time Series with lags of itself. When the residuals are autocorrelated, it means that the current value is dependent of the previous historic values and that there is a definite unexplained pattern in the Y variable that shows up in the disturbances.

The X axis corresponds to the lags of the residual, increasing in steps of 1. The very first line to the left shows the correlation of residual with itself Lag0therefore, it will always be equal to 1.

If the residuals were not autocorrelated, the correlation Y-axis from the immediate next line onwards will drop to a near zero value below the dashed blue line significance level. Clearly, this is not the case here. So we can conclude that the residuals are autocorrelated. This means there is a definite pattern in the residuals. Add lag1 of residual as an X variable to the original model.

This can be conveniently done using the slide function in DataCombine package. Unlike the acf plot of lmModthe correlation values drop below the dashed blue line from lag1 itself. Therefore we can safely assume that residuals are not autocorrelated. With a high p value of 0. So the assumption that residuals should not be autocorrelated is satisfied by this model. If, even after adding lag1 as an X variable, does not satisfy the assumption of autocorrelation of residuals, you might want to try adding lag2, or be creative in making meaningful derived explanatory variables or interaction terms.

This is more like art than an algorithm. So, the assumption holds true for this model.Linear Regression is a machine learning algorithm based on supervised learning. It performs a regression task to compute the regression coefficients. Regression models a target prediction based on independent variables.

Linear Regression performs the task to predict a dependent variable value y based on a given independent variable x. So this regression technique finds out a linear relationship between x input and y output. Hence it has got the name Linear Regression. The linear equation for univariate linear regression is given below.

The data set is shown below. We have fitted a simple linear regression model to the data after splitting the data set into train and test. The python code used to fit the data to the Linear regression algorithm is shown below. Note-theta1 is nothing but the intercept of the line and theta2 is the slope of the line.

Best fit line is a line which best fits the data which can be used for prediction. The data set which is used is the Advertising data set. This data set contains information about money spent on advertisement and their generated Sales.

## Assumptions of Multiple Linear Regression

Money was spent on TV, radio and newspaper ads. It has 3 features namely TV, radio and newspaper and 1 target Sales. There are 5 basic assumptions of Linear Regression Algorithm:. According to this assumption there is linear relationship between the features and target. Linear regression captures only linear relationship.

This can be validated by plotting a scatter plot between the features and the target. The first scatter plot of the feature TV vs Sales tells us that as the money invested on Tv advertisement increases the sales also increases linearly and the second scatter plot which is the feature Radio vs Sales also shows a partial linear relationship between them,although not completely linear.

Little or no Multicollinearity between the features:. Multicollinearity is a state of very high inter-correlations or inter-associations among the independent variables.

It is therefore a type of disturbance in the data if present weakens the statistical power of the regression model. Pair plots and heatmaps correlation matrix can be used for identifying highly correlated features. The above pair plot shows no significant relationship between the features. This heatmap gives us the correlation coefficients of each feature with respect to one another which are in turn less than 0. Why removing highly correlated features is important?

The interpretation of a regression coefficient is that it represents the mean change in the target for each unit change in an feature when you hold all of the other features constant.

The stronger the correlation, the more difficult it is to change one feature without changing another.

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It becomes difficult for the model to estimate the relationship between each feature and the target independently because the features tend to change in unison. How multicollinearity can be treated? If we have 2 features which are highly correlated we can drop one feature or combine the 2 features to form a new feature,which can further be used for prediction. Homoscedasticity Assumption:. A scatter plot of residual values vs predicted values is a goodway to check for homoscedasticity.

There should be no clear pattern in the distribution and if there is a specific pattern,the data is heteroscedastic.The regression has five key assumptions:. A note about sample size. In the software below, its really easy to conduct a regression and most of the assumptions are preloaded and interpreted for you. Screen share with a statistician as we walk you through conducting and understanding your interpreted analysis. Have your results draft complete in one hour with guaranteed accuracy. First, linear regression needs the relationship between the independent and dependent variables to be linear. It is also important to check for outliers since linear regression is sensitive to outlier effects. The linearity assumption can best be tested with scatter plots, the following two examples depict two cases, where no and little linearity is present.

Secondly, the linear regression analysis requires all variables to be multivariate normal. This assumption can best be checked with a histogram or a Q-Q-Plot. Normality can be checked with a goodness of fit test, e. When the data is not normally distributed a non-linear transformation e.

Thirdly, linear regression assumes that there is little or no multicollinearity in the data.

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Multicollinearity occurs when the independent variables are too highly correlated with each other. If multicollinearity is found in the data, centering the data that is deducting the mean of the variable from each score might help to solve the problem.

However, the simplest way to address the problem is to remove independent variables with high VIF values. Fourth, linear regression analysis requires that there is little or no autocorrelation in the data. Autocorrelation occurs when the residuals are not independent from each other. For instance, this typically occurs in stock prices, where the price is not independent from the previous price. If multicollinearity is found in the data centering the data, that is deducting the mean score might help to solve the problem. Other alternatives to tackle the problems is conducting a factor analysis and rotating the factors to insure independence of the factors in the linear regression analysis.

Fourthly, linear regression analysis requires that there is little or no autocorrelation in the data.However, before we conduct linear regression, we must first make sure that four assumptions are met:.

Linear relationship: There exists a linear relationship between the independent variable, x, and the dependent variable, y. In particular, there is no correlation between consecutive residuals in time series data.

If one or more of these assumptions are violated, then the results of our linear regression may be unreliable or even misleading. The first assumption of linear regression is that there is a linear relationship between the independent variable, x, and the independent variable, y. The easiest way to detect if this assumption is met is to create a scatter plot of x vs.

This allows you to visually see if there is a linear relationship between the two variables. If it looks like the points in the plot could fall along a straight line, then there exists some type of linear relationship between the two variables and this assumption is met. For example, the points in the plot below look like they fall on roughly a straight line, which indicates that there is a linear relationship between x and y:.

Add another independent variable to the model. For example, if the plot of x vs. The next assumption of linear regression is that the residuals are independent.

This is mostly relevant when working with time series data. The simplest way to test if this assumption is met is to look at a residual time series plot, which is a plot of residuals vs. You can also formally test if this assumption is met using the Durbin-Watson test. Depending on the nature of the way this assumption is violated, you have a few options:.

The next assumption of linear regression is that the residuals have constant variance at every level of x. When this is not the case, the residuals are said to suffer from heteroscedasticity. When heteroscedasticity is present in a regression analysis, the results of the analysis become hard to trust.

This makes it much more likely for a regression model to declare that a term in the model is statistically significant, when in fact it is not. Once you fit a regression line to a set of data, you can then create a scatterplot that shows the fitted values of the model vs. Notice how the residuals become much more spread out as the fitted values get larger. Transform the dependent variable. One common transformation is to simply take the log of the dependent variable.

For example, if we are using population size independent variable to predict the number of flower shops in a city dependent variablewe may instead try to use population size to predict the log of the number of flower shops in a city. Using the log of the dependent variable, rather than the original dependent variable, often causes heteroskedasticity to go away. Redefine the dependent variable. Use weighted regression.

This type of regression assigns a weight to each data point based on the variance of its fitted value.