Linear, nonlinear, and monotonic relationships - Minitab
A relationship of direct proportionality that, when plotted on a graph, traces a straight line. In linear relationships, any given change in an independent variable . When evaluating the relationship between two variables, it is important to determine how the variables are related. Linear relationships are most common, but. To learn what it means for two variables to exhibit a relationship that is close to linear but which contains an element of randomness. The following table gives.
Each point represents the observed x, y pair, in this case, BMI and the corresponding total cholesterol measured in each participant. Note that the independent variable BMI is on the horizontal axis and the dependent variable Total Serum Cholesterol on the vertical axis. BMI and Total Cholesterol The graph shows that there is a positive or direct association between BMI and total cholesterol; participants with lower BMI are more likely to have lower total cholesterol levels and participants with higher BMI are more likely to have higher total cholesterol levels.
For either of these relationships we could use simple linear regression analysis to estimate the equation of the line that best describes the association between the independent variable and the dependent variable. The simple linear regression equation is as follows: The Y-intercept and slope are estimated from the sample data, and they are the values that minimize the sum of the squared differences between the observed and the predicted values of the outcome, i.
These differences between observed and predicted values of the outcome are called residuals. The estimates of the Y-intercept and slope minimize the sum of the squared residuals, and are called the least squares estimates. That would mean that variability in Y could be completely explained by differences in X. Choose regression from the analysis menu after entering the data. Choose weight as a response variable and height as a predictor.
We will make frequent use of the LS fit in later chapters but there is one problem with it. It is not robust. The LS fit is easily distorted by outliers. Lets look at this using the baseball data.
Note at height 68" there is one player whose weight is at pounds. Suppose the weight was recorded as pounds.
Although high, this weight is not inconceivable for a ball player. The LS fit of this changed data is: In particular, the slope estimate has changed from 5. That is, because of one data point we now predict weight to increase 1.
We can also see the effect on the plot. Changed data LS fit Notice how the outlier pulled up the LS fit, resulting in a very poor to the bulk of the data.
One data point drove the fit! Recall that the LS fit minimizes the averaged squared deviation from the chosen line.
An outlier will have a large deviation and under the LS procedure its influence is made much greater by the squaring of this deviation. Because of the square, deviation times deviation, LS is weighing the large deviation by a large weight.
- Linear Relationship
- Linear, nonlinear, and monotonic relationships
- linear relationship
The Wilcoxon, though, uses a much smaller weight in determining the chosen line. The Wilcoxon fit is less sensitive than the LS fit at least for outliers in the Y-direction. For good data, no outliers, the Wilcoxon fit is in close agreement with the LS fit. This the Wilcoxon fit is robust fit. The regression module gives the option of a LS fit or a Wilcoxon fit. The Wilcoxon fit of the good data results in: The Wilcoxon fit can be used like the other fits for prediction.
Unlike the LS fit, the Wilcoxon fit is not sensitive to the outlier. Let X be the length cm of a laboratory mouse and let Y be its weight gm.
Linear relationship - When a changes in two variables correspond
Consider the data for X and Y given below. Obtain a scatterplot of the data and comment on the plot. For the data set in Problem 1, eyeball a linear fit obtaining an estimate of the slope and the intercept. This relationship illustrates why it is important to plot the data in order to explore any relationships that might exist. Monotonic relationship In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate.
In a linear relationship, the variables move in the same direction at a constant rate. Plot 5 shows both variables increasing concurrently, but not at the same rate.
This relationship is monotonic, but not linear.
The Pearson correlation coefficient for these data is 0. Linear relationships are also monotonic. For example, the relationship shown in Plot 1 is both monotonic and linear.