Data Means and Confidence Intervals

Figure 4-5 50% and 90% confidence intervals for the same set of random data. Filled squares are the data points whose confidence interval does not include the true population mean of 10 000. 

In analytical chemistry, the product is not spaghetti sauce, but, rather, raw data, treated data, and results. Raw data are individual values of a measured quantity, such as peak areas from a chromatogram or volumes from a buret. Treated data are concentrations or amounts found by applying a calibration procedure to the raw data. Results are what we ultimately report, such as the mean, standard deviation, and confidence interval, after applying statistics to treated data. 

Student s t-test is frequently used in statistical evaluations of environmental chemical data. It establishes a relationship between the mean (x) of normally distributed sample measurements, their sample standard deviation (,v), and the population mean (p). Confidence intervals may be calculated based on Student s t-test (Equation 10). The upper limit of the confidence interval is compared to the action level to determine whether the sampled medium contains a hazardous concentration of a pollutant. If the upper confidence limit is below the action level, the medium is not hazardous otherwise the opposite conclusion is reached. 

The interpretation of the pharmacokinetic variables Cmax, AUCs and MRT of insulin glulisine was based on 95 % confidence intervals, after ln-transformation of the data. These 95 % confidence intervals were calculated for the respective mean ratios of pair-wise treatment comparisons. In addition, the test treatment was compared to the reference treatment with respect to the pharmacokinetic variables using an ANOVA with subject, treatment and period effects, after ln-transformation of the data. The subject sum of squares was partitioned to give a term for sequence (treatment by period interaction) and a term for subject within sequence (a residual term). Due to the explorative nature of the study, no adjustment of the a-level was made for the multiple testing procedure. 

Results of iodine determined by radiochemioal neutron activation analysis for control and AD subjects are summarized in this table. Applying statistical treatment to the data sets, mean, SD, confidence interval and significance (F-test, t-test) were calculated. Where a trend is indicated to be significant the p value is <0.05. Mean values cannot be given if we have only few data (parenthetic values) therefore, statistical treatment is not possible (-). n.s. there is no significant difference between the control and AD values. 

The interval x 2s contains 95% of the sample data set, and the interval X + 3s contains 99% of the data. In practice, knowing the spread of the data about the mean is valuable, but from a practical standpoint, the interval of the mean is a confidence interval of the mean, not of the data about the mean. Fortunately, the same basic principle holds when we are interested in the standard deviation of the mean, which is s/y/n, and not the standard deviation of the data set, s. Many statisticians refer to the standard deviation of the mean as the standard error of the mean. 

Table 15.3 Mean Values and Confidence Intervals of the Animal-to-Human Permeability Coefficient Ratios for Compounds Common to the Animal and Human Data Sets...

Because simulation is a statistical tool that collects statistical data about a system or process, statistical techniques must be used to examine and analyze the data. As an example, since each run of the simulation produces one sample point and since this process involves sampling for a distribution, it is impossible to determine the relationship of a single point to the true mean. Multiple runs are required to generate enough points to determine a calculated mean, standard deviation, and confidence intervals. Then and only then should the design proceed. Knowledge and use of statistical techniques with simulation is essential to produce reliable results. 

Also, the observed delays within a replication are not independent, since applications that experience long (short) delays tend to be followed by applications that experience long (short) delays. The validity of S /N as an estimator of Var[T] rests on the assumption of i.i.d. data, and the estimator can be significantly biased when this assumption is violated. Unfortunately, the estimator is often biased low, which means that Y appears to be more precise than it actually is and confidence intervals based on S /N are inappropriately narrow. 

Fig. 7. Susceptibility of prostate cancer cells to celecoxib-induced apoptosis is independent of cyclooxygenase-2 (COX-2) expression levels. (A, left panel) Effect of 50 pM celecoxib on the viability of parental PC-3 cells and the COX-2-deficient clone 2 F6. (A, right panel) Effect of 50 pM celecoxib on the viability of the COX-2 antisense clone 7D9 with (+) or without (-) a doxycycline (Dox) pretreatment (2 pg/mL). Data represent means 95% confidence intervals (error bars) (n = 3). (B) Effect of celecoxib on the phosphorylation status of Akt and extracellular signal regulated kinase (ERK)2 in the COX-2 antisense clone 7D9 with or without doxycycline treatment (2 pg/mL) as indicated. Western blots are representatives of three independent experiments. These data indicate that the mechanism underlying celecoxib-induced apoptotic death in the 7D9 cells remained unaltered after COX-2 depletion. Source Song, X., Lin, H.P., Johnson, A.J., Tseng, P.H., Yang, Y.T., Kulp, S.K., and Chen, C.S. (2002) Cyclooxygenase-2, Player or Spectator in Cyclooxygenase-2 Inhibitor-Induced Apoptosis in Prostate Cancer Cells, j. Natl. Cancer Inst. 94, 585-591 by permission of Oxford University Press.

The mean of the difference was calculated by using the statistical hypothesis test and confidence interval estimation on the 20 participants data sample. The normality of the distribution of the sample data was tested by the Ryan—Joiner test at 5% significant level. The significance of the results with over 80% confidence level is reported in the following section. 

Having established the properties of the parameter estimates, it will now be useful to look at the predictive properties of the overall model what are the values and confidence intervals for a given prediction. Given a new data point, xa, there are two different measures for answering this question the mean response and the predicted response confidence intervals. The mean response represents the average... 

There is a temptation when analyzing data to plug numbers into an equation, carry out the calculation, and report the result. This is never a good idea, and you should develop the habit of constantly reviewing and evaluating your data. For example, if analyzing five samples gives an analyte s mean concentration as 0.67 ppm with a standard deviation of 0.64 ppm, then the 95% confidence interval is... 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

Confidence Interval for a Mean For the daily sample tensile-strength data with 4 df it is known that P[—2.132 samples exactly 16 do fall witmn the specified hmits (note that the binomial with n = 20 and p =. 90 would describe the likelihood of exactly none through 20 falling within the prescribed hmits—the sample of 20 is only a sample). 

---