# Analyzing Data with GraphPad Prism

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3 Choosing the two-way ANOVA analysis The results of two-way ANOVA Survival curves Introduction to survival curves Entering survival data Choosing a survival analysis Interpreting survival analysis Graphing survival curves Contingency tables Introduction to contingency tables Entering data into contingency tables Choosing how to analyze a contingency table Interpreting analyses of contingency tables The confidence interval of a proportion Correlation Introduction to correlation Entering data for correlation Choosing a correlation analysis Results of correlation Linear regression Introduction to linear regression Entering data for linear regression Choosing a linear regression analysis Results of linear regression Graphing linear regression Fitting linear data with nonlinear regression Introducing curve fitting and nonlinear regression Introduction to models How models are derived Why Prism can t pick a model for you Introduction to nonlinear regression Other kinds of regression Fitting a curve without choosing a model Choosing or entering an equation (model) Classic equations built-in to Prism Importing equations and the equation library User-defined equations Constraining variables in user-defined equations How to fit different portions of the data to different equations How to simulate a theoretical curve Fitting curves with nonlinear regression Fitting curves with nonlinear regression Initial values for nonlinear regression Constants for nonlinear regression Method options for nonlinear regression Output options for nonlinear regression Default options for nonlinear regression Interpreting the results of nonlinear regression Approach to interpreting nonlinear regression results Do the nonlinear regression results make sense? How certain are the best-fit values? How good is the fit? Does the curve systematically deviate from the data? Could the fit be a local minimum? Have you violated an assumption of nonlinear regression? Have you made a common error when using nonlinear regression? Comparing two curves Comparing the fits of two models Comparing fits to two sets of data (same equation) The distributions of best-fit values Why care about the distribution of best-fit values Using simulations to determine the distribution of a parameters Example simulation 1. Dose-response curves Example simulation 2. Exponential decay Detailed instructions for comparing parameter distributions Analyzing radioligand binding data Introduction to radioligand binding Law of mass action Nonspecific binding Ligand depletion Calculations with radioactivity Analyzing saturation radioligand binding data Introduction to saturation binding experiments Nonspecific binding Fitting a curve to determine B max and K d Checklist for saturation binding Scatchard plots Analyzing saturation binding with ligand depletion Analyzing competitive binding data What is a competitive binding curve? Competitive binding data with one class of receptors Shallow competitive binding curves Homologous competitive binding curves Competitive binding to two receptor types (different Kd for hot ligand) Heterologous competitive binding with ligand depletion Analyzing kinetic binding data Dissociation ("off rate") experiments Association binding experiments Analysis checklist for kinetic binding experiments Using kinetic data to test the law of mass action Kinetics of competitive binding Analyzing dose-response curves Introducing dose-response curves Fitting sigmoid dose-response curves with Prism Why Prism fits the logec50 rather than EC

14 N Critical Z 5% N Critical Z 5% Consult the references below for larger tables. You can also calculate an approximate P value as follows. 1. Calculate a t ratio from N (number of values in the sample) and Z (calculated for the suspected outlier as shown above). sampled from a Gaussian distribution. If Z is large, this P value will be very accurate. With smaller values of Z, the calculated P value may be too large The most that Grubbs' test (or any outlier test) can do is tell you that a value is unlikely to have come from the same Gaussian population as the other values in the group. You then need to decide what to do with that value. I would recommend removing significant outliers from your calculations in situations where experimental mistakes are common and biological variability is not a possibility. When removing outliers, be sure to document your decision. Others feel that you should never remove an outlier unless you noticed an experimental problem. Beware of a natural inclination to remove outliers that get in the way of the result you hope for, but to keep outliers that enhance the result you hope for. If you use nonparametric tests, outliers will affect the results very little so do not need to be removed. If you decide to remove the outlier, you then may be tempted to run Grubbs' test again to see if there is a second outlier in your data. If you do this, you cannot use the table shown above. Rosner has extended the method to detecting several outliers in one sample. See the first reference below for details. Here are two references: B Iglewicz and DC Hoaglin. How to Detect and Handle Outliers (Asqc Basic References in Quality Control, Vol 16) Amer Society for Quality Control, V Barnett, T Lewis, V Rothamsted. Outliers in Statistical Data (Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics) John Wiley & Sons, t = N( N 2) Z ( N 1) NZ 2. Determine the P value corresponding with that value of t with N-2 degrees of freedom. Use the Excel formula =TDIST(t,df,2), substituting values for t and df (the third parameter is 2, because you want a two-tailed P value). 3. Multiply the P value you obtain in step 2 by N. The result is an approximate P value for the outlier test. This P value is the chance of observing one point so far from the others if the data were all Introduction to statistical comparisons 21 Analyzing Data with GraphPad Prism 22 Copyright (c) 1999 GraphPad Software Inc.

15 Analyzing one group Frequency distributions A frequency distribution shows the distribution of Y values in each data set. The range of Y values is divided into bins and Prism determines how many values fall into each bin. Click Analyze and select built-in analyses. Then choose Frequency Distribution from the list of statistical analyses to bring up the parameters dialog. Entering data to analyze one group Prism can calculate descriptive statistics of a column of values, test whether the group mean or median differs significantly from a hypothetical value, and test whether the distribution of values differs significantly from a Gaussian distribution. Prism can also create a frequency distribution histogram. When you choose a format for the data table, choose a single column of Y values. Choose to have no X column (or choose an X column for labeling only). Enter the values for each group into a separate Y column. If your data table includes an X column, be sure to use it only for labels. Values you enter into the X column will not be analyzed by the analyses discussed here. If you format the Y columns for entry of replicate values (for example, triplicates), Prism first averages the replicates in each row. It then calculates the column statistics on the means, without considering the SD or SEM of each row. If you enter ten rows of triplicate data, the column statistics are calculated from the ten row means, not from the 30 individual values. If you format the Y columns for entry of mean and SD (or SEM) values, Prism calculates column statistics for the means and ignores the SD or SEM values you entered. Prism provides two analyses for analyzing columns of numbers. Use the frequency distribution analysis to tabulate the distribution of values in the column (and create a histogram). Use the column statistics analysis to compute descriptive statistics, to test for normality, and to compare the mean (or median) of the group against a theoretical value using a one-sample t test or the Wilcoxon rank sum test. Histograms generally look best when the bin width is a round number and there are bins. Either use the default settings, ot clear the auto option to enter the center of the first bin and the width of all bins. If you entered replicate values, Prism can either place each replicate into its appropriate bin, or average the replicates and only place the mean into a bin. All values too small to fit in the first bin are omitted from the analysis. Enter an upper limit to omit larger values from the analysis. Select Relative frequencies to determine the fraction of values in each bin, rather than the number of values in each bin. For example, if you have 50 data points of which 15 fall into the third bin, the results for the third bin will be 0.30 (15/50) rather than 15. Select Cumulative binning to see the cumulative distribution. Each bin contains the number of values that fall within or below that bin. By definition, the last bin contains the total number of values. Analyzing one group 23 Analyzing Data with GraphPad Prism 24 Copyright (c) 1999 GraphPad Software Inc.

16 Column statistics From your data table, click Analyze and choose built-in analyses. Then choose Column Statistics from the list of statistical analyses to bring up the parameters dialog. To interpret the results, see "Interpreting descriptive statistics" on page 26,"The results of normality tests" on page 29, "The results of a one-sample t test" on page 30, or "The results of a Wilcoxon rank sum test" on page 34. Interpreting descriptive statistics SD and CV The standard deviation (SD) quantifies variability. If the data follow a bellshaped Gaussian distribution, then 68% of the values lie within one SD of the mean (on either side) and 95% of the values lie within two SD of the mean. The SD is expressed in the same units as your data. Prism calculates the SD using the equation below. (Each y i is a value, ymean is the average, and N is sample size). SD = ( yi ymean) 2 N 1 Choose the descriptive statistics you want to determine. Note that CI means confidence interval. Check "test whether the distribution is Gaussian" to perform a normality test. To determine whether the mean of each column differs from some hypothetical value, choose the one-sample t test, and enter the hypothetical value, which must come from outside this experiment. The hypothetical value could be a value that indicates no change such as 0.0, 1.0 or 100, or it could come from theory or previous experiments. The one-sample t test assumes that the data are sampled from a Gaussian distribution. The Wilcoxon rank sum test compares whether the median of each column differs from a hypothetical value. It does not assume that the distribution is Gaussian, but generally has less power than the one-sample t test. Choosing between a one-sample t test and a Wilcoxon rank sum test involves similar considerations to choosing between an unpaired t test and the Mann-Whitney nonparametric test. See "t test or nonparametric test?" on page 42. The standard deviation computed this way (with a denominator of N-1) is called the sample SD, in contrast to the population SD which would have a denominator of N. Why is the denominator N-1 rather than N? In the numerator, you compute the difference between each value and the mean of those values. You don t know the true mean of the population; all you know is the mean of your sample. Except for the rare cases where the sample mean happens to equal the population mean, the data will be closer to the sample mean than it will be to the population mean. This means that the numerator will be too small. So the denominator is reduced as well. It is reduced to N-1 because that is the number of degrees of freedom in your data. There are N-1 degrees of freedom, because you could calculate the remaining value from N-1 of the values and the sample mean. The coefficient of variation (CV) equals the standard deviation divided by the mean (expressed as a percent). Because it is a unitless ratio, you can compare the CV of variables expressed in different units. It only makes sense to report CV for variables where zero really means zero, such as mass or enzyme activity. Don't calculate CV for variables, such as temperature, where the definition of zero is somewhat arbitrary. SEM The standard error of the mean (SEM) is a measure of how far your sample mean is likely to be from the true population mean. The SEM is calculated by this equation: Analyzing one group 25 Analyzing Data with GraphPad Prism 26 Copyright (c) 1999 GraphPad Software Inc.

17 SEM = SD N With large samples, the SEM is always small. By itself, the SEM is difficult to interpret. It is easier to interpret the 95% confidence interval, which is calculated from the SEM. 95% Confidence interval The confidence interval quantifies the precision of the mean. The mean you calculate from your sample of data points depends on which values you happened to sample. Therefore, the mean you calculate is unlikely to equal the overall population mean exactly. The size of the likely discrepancy depends on the variability of the values (expressed as the SD) and the sample size. Combine those together to calculate a 95% confidence interval (95% CI), which is a range of values. You can be 95% sure that this interval contains the true population mean. More precisely, if you generate many 95% CIs from many data sets, you expect the CI to include the true population mean in 95% of the cases and not to include the true mean value in the other 5% of the cases. Since you don't know the population mean, you'll never know when this happens. The confidence interval extends in each direction by a distance calculated from the standard error of the mean multiplied by a critical value from the t distribution. This value depends on the degree of confidence you want (traditionally 95%, but it is possible to calculate intervals for any degree of confidence) and on the number of degrees of freedom in this experiment (N-1). With large samples, this multiplier equals With smaller samples, the multiplier is larger. Quartiles and range Quartiles divide the data into four groups, each containing an equal number of values. Quartiles divided by the 25 th percentile, 50 th percentile, and 75 th percentile. One quarter of the values are less than or equal to the 25 th percentile. Three quarters of the values are less than or equal to the 75 th percentile. The median is the 50 th percentile. Prism computes percentile values by first computing R= P*(N+1)/100, where P is 25, 50 or 75 and N is the number of values in the data set. The result is the rank that corresponds to the percentile value. If there are 68 values, the 25 th percentile corresponds to a rank equal to.25*69= So the 25 th percentile lies between the value of the 17 th and 18 th value (when ranked from low to high). Prism computes the 25 th percentile as the average of those two values (some programs report the value one quarter of the way between the two). Analyzing one group 27 Geometric mean The geometric mean is the antilog of the mean of the logarithm of the values. It is less affected by outliers than the mean. Which error bar should you choose? It is easy to be confused about the difference between the standard deviation (SD) and standard error of the mean (SEM). The SD quantifies scatter how much the values vary from one another. The SEM quantifies how accurately you know the true mean of the population. The SEM gets smaller as your samples get larger. This makes sense, because the mean of a large sample is likely to be closer to the true population mean than is the mean of a small sample. The SD does not change predictably as you acquire more data. The SD quantifies the scatter of the data, and increasing the size of the sample does not increase the scatter. The SD might go up or it might go down. You can't predict. On average, the SD will stay the same as sample size gets larger. If the scatter is caused by biological variability, your probably will want to show the variation. In this case, graph the SD rather than the SEM. You could also instruct Prism to graph the range, with error bars extending from the smallest to largest value. Also consider graphing every value, rather than using error bars. If you are using an in vitro system with no biological variability, the scatter can only result from experimental imprecision. In this case, you may not want to show the scatter, but instead show how well you have assessed the mean. Graph the mean and SEM or the mean with 95% confidence intervals. Ideally, the choice of which error bar to show depends on the source of the variability and the point of the experiment. In fact, many scientists always show the mean and SEM, to make the error bars as small as possible. Other results For other results, see "The results of normality tests" on page 29, "The results of a one-sample t test" on page 30, or "The results of a Wilcoxon rank sum test" on page 34. The results of normality tests How the normality test works Prism tests for deviations from Gaussian distribution using the Kolmogorov- Smirnov (KS) test. Since the Gaussian distribution is also called the Normal Analyzing Data with GraphPad Prism 28 Copyright (c) 1999 GraphPad Software Inc.

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