78 introduction to statistics for bioMEDical engineers
and
b = y − mx,
where
– |
1 |
N-1 |
– |
1 |
N-1 |
x = |
N— Σxi and |
y |
= N—Σyi . |
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i=0 |
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i=0 |
Hence, once we have our measured data, we can simply use our equations for m and b to find the line, or linear model, of best fit.
The Correlation Coefficient
It is important to realize that linear regression will fit a line to any two sets of data regardless of how well the data are modeled by a linear model. Even if the data, when plotted as a scatterplot, look nothing like a line, linear regression will fit a line to the data. As biomedical engineers, we have to ask, “How well does the measured data “fit” the line estimated through linear regression?”
One measure of how well the experimental data fit the linear model is the correlation coefficient. The correlation coefficient, r, has a value between −1 and 1 and indicates how well the linear model fits to the data.
The correlation coefficient, r, may be estimated from the experimental data, xi and yi, using the following equation:
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N −1 |
(xi − x )(yi − y ) |
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r = |
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Σ |
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i =0 |
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, |
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[ |
N −1 |
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N −1 |
(yi − y )2 |
1/ 2 |
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Σ |
(xi − x )2 Σ |
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i =0 |
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i =0 |
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] |
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where |
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– |
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1 N-1 |
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– |
1 N-1 |
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x = |
N— |
Σxi and |
y |
= N—Σyi . |
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i=0 |
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i=0 |
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It is important to note that an r = 0 does not mean that the two processes, x and y, are independent. It simply indicates that any dependency between x and y is not well described or modeled by a linear relation. There could be a nonlinear relation between x and y. An r = 0 simply means
Linear Regression and Correlation Analysis 79
that x and y are uncorrelated in a linear sense. That is, one may not predict y from x using a linear model, y = mx + b.
A measure related to the correlation coefficient, r, is the coefficient of determination, R2, which is a summary statistic that tells us how well our regression model fits our data. R2 can be used as measure of goodness of fit for any regression model, not just linear regression. For linear regression, R2 is the square of the correlation coefficient and has a value between 0 and 1. The coefficient of determination tells us how much of the variability in the data may be explained by the model parameters as a fraction of total variability in the data.
It is important to realize that the estimated slope of best fit and the correlation coefficient are statistics that may or may not be significant. Thus, t tests may be performed to test if the slope estimated through linear is significantly different from zero [3]. Likewise, t tests may be performed to test if the correlation coefficient is significantly different from zero. Finally, we may also compute confidence intervals for the estimated slope [3].
• • • •
81
c h a p t e r 7
Power Analysis and Sample Size
Up to this point, we have discussed important aspects of experimental design, data summary, and statistical analysis that will allow us to test hypotheses and draw conclusions with some level of confidence.
However, we have not yet addressed a very important question. The question we ask now is, “how large should my sample be to capture the variability in my underlying population so that my types I and II error rates will be small?” In other words, how large of a sample is required such that the probability of making a type I or II error in rejecting or accepting a null hypothesis will be acceptable under the circumstances. Different situations call for different error rates. An error such as diagnosing streptococcal throat infection when streptococcus bacteria are not present is likely not as serious as missing the diagnosis of cancer. Another way of phrasing the question is, “How powerful is my statistical analysis in accepting or rejecting the null hypothesis?”
If the sample size is too small, the consequence may be that we miss an effect (type II error) because we do not have enough power in the test to demonstrate, with confidence, an effect.
However, when choosing a sample size, it is too easy to simply say that the sample size should be as large as possible. Even if an investigator had access to as many samples as he or she desired, there are practical considerations and constraints that limit the sample size. If the sample size is too large, there are economic and ethical problems to consider. First, there are expenses associated with running an experiment, such as a clinical trial. There are costs associated with the personnel who run the experiments, the experimental units (animals, cell cultures, compensation for human time), perhaps drugs and other medical procedures that are administered, and others. Thus, the greater the number of samples, the greater the expense. Clinical trials are typically very expensive to run.
The second consideration for limiting sample size is an ethical concern. Many biomedicalrelated experiments or trials involve human or animal subjects. These subjects may be exposed to experimental drugs or therapies that involve some risk, and in the case of animal studies, the animal may be sacrificed at the end of an experiment. Bottom line, we do not wish to use human or animal subjects for no good reason, especially if we gain nothing in terms of the power of our statistical analysis by increasing the sample size.