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>>>y = ax + b (y = 82.9842x + 23.5645)

>>>x = 2 y = 189.533

The first one showing the line equation, the second the value predicted for the quantity of items sold when the unit cost is $$ 2,00 and finally, the third line brings the evaluated Rˆ2 metrics.

Other important observation is that we cannot forget that this generated equation not necessarily provide all the scatters of the plot, that is, by using the equation we can not obtain exactly the same values of the previously data, thus the equation generated by the linear regression creates an approximation of the values.

BSCI 1510L Literature and Stats Guide: 6 Scatter plot, trendline, and linear regression

A regression analysis can provide three forms of descriptive information about the data included in the analysis: the equation of the best fit line, an R2 value, and a P-value.

A regression analysis of these data calculates that the equation of the best fit line is y = 6x + 55 .

Right: R2=0.16, P=0.029 There are other patterns of data for which the best fit line would also be y = 6x + 55 .

Its value ranges from 0 (essentially a random cloud of points) to 1 (the points fall perfectly on a straight line).

An R2 of 0.94 means that 94% of the variance in the data is explained by the line and 6% of the variance is due to unexplained effects.

The value of P is accordingly high, indicating that it is probable that the slope could deviate from zero by this amount based solely on chance.

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11. Correlation and regression

The word correlation is used in everyday life to denote some form of association.

If one set of observations consists of experimental results and the other consists of a time scale or observed classification of some kind, it is usual to put the experimental results on the vertical axis.

It is reasonable, for instance, to think of the height of children as dependent on age rather than the converse but consider a positive correlation between mean tar yield and nicotine yield of certain brands of cigarette.' The nicotine liberated is unlikely to have its origin in the tar: both vary in parallel with some other factor or factors in the composition of the cigarettes.

As a further example, a plot of monthly deaths from heart disease against monthly sales of ice cream would show a negative association.

It is simply that the mortality rate from heart disease is inversely related - and ice cream consumption positively related - to a third factor, namely environmental temperature.

The data are given in table 11.1 and the scatter diagram shown in figure 11.2 Each dot represents one child, and it is placed at the point corresponding to the measurement of the height (horizontal axis) and the dead space (vertical axis).

The registrar now inspects the pattern to see whether it seems likely that the area covered by the dots centres on a straight line or whether a curved line is needed.

When making the scatter diagram (figure 11.2 ) to show the heights and pulmonary anatomical dead spaces in the 15 children, the paediatrician set out figures as in columns (1), (2), and (3) of table 11.1 .

The calculation of the correlation coefficient is as follows, with x representing the values of the independent variable (in this case height) and y representing the values of the dependent variable (in this case anatomical dead space).

The correlation coefficient of 0.846 indicates a strong positive correlation between size of pulmonary anatomical dead space and height of child.

part of the variation in one of the variables (as measured by its variance) can be thought of as being due to its relationship with the other variable and another part as due to undetermined (often 'random') causes.

If we wish to label the strength of the association, for absolute values of r, 0-0.19 is regarded as very weak, 0.2-0.39 as weak, 0.40-0.59 as moderate, 0.6-0.79 as strong and 0.8-1 as very strong correlation, but these are rather arbitrary limits, and the context of the results should be considered.

Entering table B at 15 - 2 = 13 degrees of freedom we find that at t = 5.72, P<0.001 so the correlation coefficient may be regarded as highly significant.

The assumptions governing this test are: The test should not be used for comparing two methods of measuring the same quantity, such as two methods of measuring peak expiratory flow rate.

The regression equation representing how much y changes with any given change of x can be used to construct a regression line on a scatter diagram, and in the simplest case this is assumed to be a straight line.

(the regression coefficient) signifies the amount by which change in x must be multiplied to give the corresponding average change in y, or the amount y changes for a unit increase in x.

It can easily be shown that any straight line passing through the mean values x and y will give a total prediction error of zero because the positive and negative terms exactly cancel.

To remove the negative signs we square the differences and the regression equation chosen to minimise the sum of squares of the prediction errors, We denote the sample estimates of Alpha and Beta by a and b.

The way to draw the line is to take three values of x, one on the left side of the scatter diagram, one in the middle and one on the right, and substitute these in the equation, as follows: If x = 110, y = (1.033 x 110) - 82.4 = 31.2 If x = 140, y = (1.033 x 140) - 82.4 = 62.2 If x = 170, y = (1.033 x 170) - 82.4 = 93.2 Although two points are enough to define the line, three are better as a check.

Figure 11.3 Regression line drawn on scatter diagram relating height and pulmonaiy anatomical dead space in 15 children The standard error of the slope SE(b) is given by:

They show how one variable changes on average with another, and they can be used to find out what one variable is likely to be when we know the other - provided that we ask this question within the limits of the scatter diagram.

For instance, a regression line might be drawn relating the chronological age of some children to their bone age, and it might be a straight line between, say, the ages of 5 and 10 years, but to project it up to the age of 30 would clearly lead to error.

to allow for covariates - in a clinical trial the dependent variable may be outcome after treatment, the first independent variable can be binary, 0 for placebo and 1 for active treatment and the second independent variable may be a baseline variable, measured before treatment, but likely to affect outcome.

Having obtained the regression equation, calculate the residuals A histogram of will reveal departures from Normality and a plot of versus will reveal whether the residuals increase in size as increases.

11.3 If the values of x from the data in 11.1 represent mean distance of the area from the hospital and values of y represent attendance rates, what is the equation for the regression of y on x?

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