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n n n n n Advantages of Multivariate Analysis Close resemblance to how the researcher thinks. Easy visualisation and interpretation of data. More information is analysed simultaneously, giving greater power. Relationship between variables is understood better. Focus shifts from individual factors taken singly to relationship among variables. Definitions - I n n n n Independent (or Explanatory or Predictor) variable always on the X axis. Dependent (or Outcome or Response) variable always on the Y axis. In OBSERVATIONAL studies researcher observes the effects of explanatory variables on outcome. In INTERVENTION studies researcher manipulates explanatory variable (e.g. dose of drug) to influence outcome Definitions - II n Scatter plot helps to visualise the relationship between two variables. The figure shows a scatter plot with a regression line. For a given value of X there is a spread of Y values. The regression line represents the mean values of Y. Scatterplot of deuterium against testweighing Deuterium = -67.3413 + 1.16186 Test weigh S = 234.234 R-Sq = 59.3 % R-Sq(adj) = 56.0 % 2500 Deuterium n 2000 1500 Regression 95% CI 1000 1000 1500 Test weigh 2000 Definitions - III n n INTERCEPT is the value of Y for X = 0. It denotes the point where the regression line meets the Y axis SLOPE is a measure of the change in the value of Y for a unit change in X. Y axis Slope Intercept X axis Basic Assumptions n n n n Y increases or decreases linearly with increase or decrease in X. For any given value of X the values of Y are distributed Normally. Variance of Y at any given value of X is the same for all value of X. The deviations in any one value of Y has no effect on other values of Y for any given X The Residuals n n The difference between the observed value of Y and the value on the regression line (Fitted value) is the residual. The statistical programme minimizes the sum of the squares of the residuals. In a Good Fit the data points are all crowded around the regression line. Residual Analysis of Variance - I n n n The variation of Y values around the regression line is a measure of how X and Y relate to each other. Method of quantifying the variation is by Analysis of variance presented as Analysis of Variance table Total sum of squares represents total variation of Y values around their mean - Syy Analysis of Variance - II Total Sum of Squares ( Syy ) is made up of two parts: (i). Explained by the regression (ii). Residual Sum of Squares Sum of Squares ÷ its degree of freedom = Mean Sum of Squares (MSS) The ratio MSS due to regression ÷ MSS Residual = F ratio Reading the output n n n n n n n n n n Regression Equation Residual Sum of Squares (RSS) Values of α and β. R2 S (standard deviation) Testing for β ≠ 0 Analysis of Variance Table F test Outliers Remote from the rest