REGRESSION AND CORRELATION ANALYSIS

Transcription

1 Problem 1 Problem 2 A group of 625 students has a mean age of 15.8 years with a standard devia>on of 0.6 years. The ages are normally distributed. How many students are younger than 16.2 years?

2 REGRESSION AND CORRELATION ANALYSIS

3 Regression analysis: From experimental or theore>cal work we know the func>on- like rela>onship between one or more variables, but the constants in equa>ons are not necessarily known. Regression analysis provides techniques to determine the constants with a given reliability intervals. Correla:on analysis: This sta>s>cal method determines a constant (correla>on coefficient) that measures the strength of the rela>onship between variables; this method is the correla>on analysis.

4 It is possible that there is no lawful connec>on between the variables, moreover, the independent variable is not even con>nuous. Also in this case we can try to set up a func>on- like rela>on between the variables, which is the regression func3on or equa3on. It is important to note that a significant regression rela>on between the variables does not necessarily mean a causal rela3onship;

5 The ra>onale in this correla>on is that small shoe size (English unit of measurement) may indicate a possible birth difficulty due to a small pelvis. The analysis of data gives significant results, as there is strong evidence of a trend between the data (Figure 12.2). This rela>on is not directly causal, trend is a convenient indicator for doctors that may occasionally indicate a problem.

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7 Linear regression A special but at the same >me important case of regression analysis occurs when there is a linear rela>on between one or more variables. The graphical image of linear rela>on in case of two variables is a straight line, or regression line, in case of more variables we talk about regression plane. The corresponding value pairs of two non- independent random variables(x, Y) should be: If we label the independent variable X, and the dependent variable Y, then the equa>on characterising the linear rela>onship is the following: where a and b are the already known constants (a is the intercept, b is the slope of straight line), E is a random variable, its mean is 0. E is the variable error that represents a part of Y, which cannot be explained by X.

8 A legkisebb négyzetek módszere The best possible fit is if we choose the posi>on when the sum of differences around the line Y is minimal, meaning that the difference that cannot be explained by regression should be the smallest possible.

9 The equa>on of the line: By choosing the a and b parameters well we are able to reach that the sum of squares of random devia>ons be minimal about the Y = a + b x line: As a result of the mathema>cal method the slope of the line is: is called the regression equa3on of variable Y regarding variable X.

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11 Standard deviáció számítása In order to calculate the standard deviance of b regression coefficient, let s compare the dispersion of yi about with the dispersion of the points of the regression line (Yi) about

12 The standard error of regression es3mate is given by the residual sum of squares, the variability unexplained by regression:

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14 A regressziós együ[hatóra vonatkozó hipotézisvizsgálat The devia>on unexplained by regression in most cases can be a[ributed to random error, therefore, we can assume that the y i measurement result for different x i shows normal distribu>on about the real or popula3on regression line, and the variance does not depend on x. Any func>on of a random variable is a random variable, therefore a, b and Y are also random variables. It can be proved that the regression coefficient b is a normally distributed random variable, thus the b/se(b) random variable follows a t- distribu3on with (n- 2) degrees of freedom. With the help of the null hypothesis for the popula>on regression coefficient (β) we can analyse whether the rela>on between the two variables is a real correla>on or seemingly real.

15 1. T- teszt Null hypothesis: the devia>on of the popula>on s βregression coefficient from zero is due to random effects. Alterna3ve hypothesis: theβregression coefficient refers to the real rela>on between the two variables. Since b can be both posi>ve and nega>ve, the probability has to be checked on both ends of the distribu>on (two- sided hypothesis tes3ng).

16 2. ANOVA

17 Null hypothesis: the two variances are from the same popula>on, the variance explained by regression and the residual variance may differ from each other due to random effects. Alterna3ve hypothesis: the two variances are from different popula>ons, the rela>on between the two variables is a real correla>on.

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20 Null hypothesis: there is no rela>on between the dose of an>coagulant and the prothrombin >me. An equal defini>on to this: the coefficient of popula>on regression line is zero: H0: β = 0 Alterna3ve hypothesis: the prothrombin >me linearly depends on the an>coagulant concentra>on that means the coefficient of popula>on regression line is not zero: H1: β 0 We can use both the t- test and the F- test for hypothesis tes>ng. The test sta>s>c for the regression coefficient is:

21 At a P = 0.05 probability level in case of variances with f1 = 1and f2 = 10 degrees of freedom the table for the F sta>s>c: At a P = 0.05 probability level we refuse the null hypothesis and accept the alterna>ve hypothesis, meaning that the two variance cannot come from the same popula>on. It means again that there is a real correla>on between the two variables.

22 Least squares principle The method provided by the least squares principle is oeen applied in such measurement procedures where physical, chemical or biological data pairs are measured, and it is assumed that there is a known func>onal rela>onship between the observed variables. The parameters of the selected func>on can be determined by minimalizing the sum of squared devia>ons of the observed data and the data fi[ed with the help of the func>on. In the form of an equa>on it is: where n is the number of the data pairs, Y i is the value of the func>on for the(x i, y i )data pair. The summa>on includes all data pairs. The least squares principle can be applied to such y = f(x,α,β,γ) model func>ons where - (α, β, γ) are the parameters of the func>on that fulfil the following condi>ons: (1) the distribu>on of variable y approximates normal distribu>on about the observed Y value of the real model func>on for any values of variable x; (2) the variance of variable y does not depend on the selec>on of x, meaning that the precision of determining variable y does not depend on x;

23 Logis>c regression (dose- response problem) The main concern of the medical treatment of diseases is how the pa>ents suffering from the same disease react to the treatment with the same medica>on. It is obvious that due to the biological variability the response to the dose is different; some pa>ents exhibit the same response for lower doses while others for higher dose.

24 Test for checking the linearity of regression In a case where regression is considered the first thing we assume between the variables X and Y is a linear rela>onship, and then try to do the analysis based on the linear model. This model is oeen not adequate for solving the problem. A quick and simple model to check linearity is based on hypothesis tes>ng. The method analyzes the randomness of the sign sequence differences of the serialised yi Yi. If it is a random sample, then the linear approach is effec>ve.

25 The average systolic blood pressure of young females ages between 8 and 20 follows non- linear rela>on. Analysing the yi Yi differences, amongst the higher age values the differences appear to be nega>ve. A more detailed observa>on would show that the linear model is not suitable for analysis in this age group. An effec>ve choice would be the parabolic quadric surface fihng.