Bonferroni interval estimates

Bonferroni Interval Estimates Interval estimates for the unknown X, referred to as unlimited simultaneous discrimination intervals ( 21 ), are based on the estimated regression line of y, on X., and the confidence interval (on the Y-axis) about the response y for an unknown The resulting interval estimates have the property that for at least 100(1- )% of the different calibration sets, at least 100P% of the amount intervals estimated from that calibration will contain true unknown amounts... 

The Bonferroni interval estimate of X, given Y, is found in three moves. First, the Working-Hotelling confidence band for the regression line... 

Bonferroni inequality is invoked to combine the two proceeding confidence statements, each made with the confidence (l-a/2), to yield an interval estimate for X with confidence at least (1-a). The confidence band on the regression line and the confidence interval on U are intersected and the Bonferroni interval estimate of X is found by projecting the intersection onto the x-axis. Figure Ic illustrates the procedure. If is in the interval on the Y-axis and if the hyperbolic confidence band contains the line... 

The technique for obtaining interval estimates for X, discussed in this section, is presented in the paper by Lieberman, Miller, and Hamilton ( 2 ) and based on the Bonferroni inequality ( ) described below Other methods are found in the references ( 23,24 ) ... 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

Often we have data from several populations that we believe follow the same parametric distribution (such as the normal distribution), but may have different values of the parameter (such as the mean). The classical frequentist approach would be to analyze each population separately. The maximum likelihood estimate of the parameter for each population would be estimated from the sample from that population. Simultaneous confidence intervals such as Bonferroni, Tiikey, or Scheff6 intervals would be used for the difference between different population parameter values. These wider intervals would control the overall confidence level, and the overall significance level for testing the hypothesis that the differences between all the population parameters are zero. However, these intervals don t do anything about the parameter estimates themselves. 

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