# Regression And Correlation Analysis Examples Pdf

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*In many studies, we measure more than one variable for each individual. For example, we measure precipitation and plant growth, or number of young with nesting habitat, or soil erosion and volume of water.*

- Pearson Correlation and Linear Regression
- Chapter 7: Correlation and Simple Linear Regression
- Statistics review 7: Correlation and regression
- Multivariate Regression Analysis | Stata Data Analysis Examples

*Linear regression is the next step up after correlation.*

The present review introduces methods of analyzing the relationship between two quantitative variables. The calculation and interpretation of the sample product moment correlation coefficient and the linear regression equation are discussed and illustrated. Common misuses of the techniques are considered.

## Pearson Correlation and Linear Regression

In many studies, we measure more than one variable for each individual. For example, we measure precipitation and plant growth, or number of young with nesting habitat, or soil erosion and volume of water.

We collect pairs of data and instead of examining each variable separately univariate data , we want to find ways to describe bivariate data , in which two variables are measured on each subject in our sample.

Given such data, we begin by determining if there is a relationship between these two variables. As the values of one variable change, do we see corresponding changes in the other variable? We can describe the relationship between these two variables graphically and numerically.

We begin by considering the concept of correlation. Correlation is defined as the statistical association between two variables. A correlation exists between two variables when one of them is related to the other in some way. A scatterplot is the best place to start. A scatterplot or scatter diagram is a graph of the paired x, y sample data with a horizontal x-axis and a vertical y-axis.

Each individual x, y pair is plotted as a single point. In this example, we plot bear chest girth y against bear length x. When examining a scatterplot, we should study the overall pattern of the plotted points. In this example, we see that the value for chest girth does tend to increase as the value of length increases. We can see an upward slope and a straight-line pattern in the plotted data points. Linear relationships can be either positive or negative.

Positive relationships have points that incline upwards to the right. As x values increase, y values increase. As x values decrease, y values decrease. For example, when studying plants, height typically increases as diameter increases. Negative relationships have points that decline downward to the right. As x values increase, y values decrease. As x values decrease, y values increase. For example, as wind speed increases, wind chill temperature decreases.

Non-linear relationships have an apparent pattern, just not linear. For example, as age increases height increases up to a point then levels off after reaching a maximum height.

When two variables have no relationship, there is no straight-line relationship or non-linear relationship. When one variable changes, it does not influence the other variable. Because visual examinations are largely subjective, we need a more precise and objective measure to define the correlation between the two variables.

To quantify the strength and direction of the relationship between two variables, we use the linear correlation coefficient:. The sample size is n. This statistic numerically describes how strong the straight-line or linear relationship is between the two variables and the direction, positive or negative. Correlation is not causation!!! Just because two variables are correlated does not mean that one variable causes another variable to change.

Examine these next two scatterplots. Plot 1 shows little linear relationship between x and y variables. Plot 2 shows a strong non-linear relationship. Ignoring the scatterplot could result in a serious mistake when describing the relationship between two variables. When you investigate the relationship between two variables, always begin with a scatterplot.

This graph allows you to look for patterns both linear and non-linear. Once you have established that a linear relationship exists, you can take the next step in model building. Once we have identified two variables that are correlated, we would like to model this relationship. We want to use one variable as a predictor or explanatory variable to explain the other variable, the response or dependent variable.

In order to do this, we need a good relationship between our two variables. The model can then be used to predict changes in our response variable. A strong relationship between the predictor variable and the response variable leads to a good model. A simple linear regression model is a mathematical equation that allows us to predict a response for a given predictor value.

The slope describes the change in y for each one unit change in x. A hydrologist creates a model to predict the volume flow for a stream at a bridge crossing with a predictor variable of daily rainfall in inches. The y-intercept of 1.

The slope tells us that if it rained one inch that day the flow in the stream would increase by an additional 29 gal. If it rained 2 inches that day, the flow would increase by an additional 58 gal. The Least-Squares Regression Line shortcut equations. An alternate computational equation for slope is:. This simple model is the line of best fit for our sample data.

The regression line does not go through every point; instead it balances the difference between all data points and the straight-line model. The difference between the observed data value and the predicted value the value on the straight line is the error or residual.

The criterion to determine the line that best describes the relation between two variables is based on the residuals. For example, if you wanted to predict the chest girth of a black bear given its weight, you could use the following model. But a measured bear chest girth observed value for a bear that weighed lb.

A negative residual indicates that the model is over-predicting. A positive residual indicates that the model is under-predicting. In this instance, the model over-predicted the chest girth of a bear that actually weighed lb. This random error residual takes into account all unpredictable and unknown factors that are not included in the model.

An ordinary least squares regression line minimizes the sum of the squared errors between the observed and predicted values to create a best fitting line. The differences between the observed and predicted values are squared to deal with the positive and negative differences.

After we fit our regression line compute b 0 and b 1 , we usually wish to know how well the model fits our data. To determine this, we need to think back to the idea of analysis of variance. In ANOVA, we partitioned the variation using sums of squares so we could identify a treatment effect opposed to random variation that occurred in our data.

The idea is the same for regression. We want to partition the total variability into two parts: the variation due to the regression and the variation due to random error. And we are again going to compute sums of squares to help us do this. Suppose the total variability in the sample measurements about the sample mean is denoted by , called the sums of squares of total variability about the mean SST. The squared difference between the predicted value and the sample mean is denoted by , called the sums of squares due to regression SSR.

The SSR represents the variability explained by the regression line. Finally, the variability which cannot be explained by the regression line is called the sums of squares due to error SSE and is denoted by. SSE is actually the squared residual. The sums of squares and mean sums of squares just like ANOVA are typically presented in the regression analysis of variance table.

The ratio of the mean sums of squares for the regression MSR and mean sums of squares for error MSE form an F-test statistic used to test the regression model. The larger the explained variation, the better the model is at prediction. The larger the unexplained variation, the worse the model is at prediction. A quantitative measure of the explanatory power of a model is R 2 , the Coefficient of Determination:. The Coefficient of Determination measures the percent variation in the response variable y that is explained by the model.

The Coefficient of Determination and the linear correlation coefficient are related mathematically. Even though you have determined, using a scatterplot, correlation coefficient and R 2 , that x is useful in predicting the value of y , the results of a regression analysis are valid only when the data satisfy the necessary regression assumptions. We can use residual plots to check for a constant variance, as well as to make sure that the linear model is in fact adequate.

The center horizontal axis is set at zero. One property of the residuals is that they sum to zero and have a mean of zero. A residual plot should be free of any patterns and the residuals should appear as a random scatter of points about zero.

A residual plot with no appearance of any patterns indicates that the model assumptions are satisfied for these data. The residuals tend to fan out or fan in as error variance increases or decreases.

The model may need higher-order terms of x , or a non-linear model may be needed to better describe the relationship between y and x. Transformations on x or y may also be considered. A normal probability plot allows us to check that the errors are normally distributed.

It plots the residuals against the expected value of the residual as if it had come from a normal distribution. Recall that when the residuals are normally distributed, they will follow a straight-line pattern, sloping upward. The most serious violations of normality usually appear in the tails of the distribution because this is where the normal distribution differs most from other types of distributions with a similar mean and spread. Curvature in either or both ends of a normal probability plot is indicative of nonnormality.

## Chapter 7: Correlation and Simple Linear Regression

As the name implies, multivariate regression is a technique that estimates a single regression model with more than one outcome variable. When there is more than one predictor variable in a multivariate regression model, the model is a multivariate multiple regression. Please Note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics and potential follow-up analyses. Example 1. A researcher has collected data on three psychological variables, four academic variables standardized test scores , and the type of educational program the student is in for high school students.

Expand your knowledge. Your time is valuable. Cut through the noise and dive deep on a specific topic with one of our curated content hubs. Interested in engaging with the team at G2? Check it out and get in touch! It all comes down to correlation and regression, which are statistical analysis measurements used to find connections between two variables, measure the connections, and make predictions. Measuring correlation and regression is commonly used in a variety of industries, and it can also be seen in our daily lives.

## Statistics review 7: Correlation and regression

This web book is composed of three chapters covering a variety of topics about using SPSS for regression. We should emphasize that this book is about "data analysis" and that it demonstrates how SPSS can be used for regression analysis, as opposed to a book that covers the statistical basis of multiple regression. We assume that you have had at least one statistics course covering regression analysis and that you have a regression book that you can use as a reference see the Regression With SPSS page and our Statistics Books for Loan page for recommended regression analysis books.

A correlation or simple linear regression analysis can determine if two numeric variables are significantly linearly related. A correlation analysis provides information on the strength and direction of the linear relationship between two variables, while a simple linear regression analysis estimates parameters in a linear equation that can be used to predict values of one variable based on the other. The Pearson correlation coefficient, r , can take on values between -1 and 1. A general form of this equation is shown below:. The slope, b 1 , is the average change in Y for every one unit increase in X.

*Linear regression models. Notes on linear regression analysis pdf.*

### Multivariate Regression Analysis | Stata Data Analysis Examples

The objective of many statistical analysis is to make predictions. For example, in canola cultivation it may be of interest to predict the canola crop yield the dependent or response variable for different levels of nitrogen fertilizer the independent or explanatory variable. Such prediction require to find a mathematical formula a statistical model which relates the dependent variable to one or more independent variables. In countless real-world problems such relationship is not deterministic: it must be a random component to the formula that relates the variables.

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Answer: REGRESSION ANALYSIS. Which derives a description of the functional nature of the relationship between two or more variables. Examples.

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*В тот год аналогичное приглашение получили еще сорок кандидатов.*