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Introduction Statistics are increasingly prevalent in medical practice, and for those doing research, statistical issues are fundamental. It is extremely important therefore, to understand basic statistical ideas relating to research design and data analysis, and to be familiar with the most commonly used methods of analysis.

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Although data analysis is certainly an important part of the statistical process, there is an equally vital role to be played in the design of the research project. Without a properly designed study, the subsequent analysis may be unsafe, and/or a complete waste of time and resources.

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Types of data Descriptive statistics Data distributions Comparative statistics Non-parametric tests Paired data Comparison of several means Comparing proportions Exploring the relationship between 2 variables Correlation Linear regression Survival analysis

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Proportion of total Platelet count 010040010001500 0.05.1.15 900

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Types of Data Categorical –binary or dichotomous e.g. diabetic/non-diabetic, smoker/non-smoker –nominal e.g. AB/B/AB/O, short-sighted/long- sighted/normal –ordered categorical (ordinal) e.g. stage 1/2/3/4, mild/moderate/severe

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Discrete numerical - e.g. number of children - 0/1/2/3/4/5+ Continuous - e.g. Blood pressure, age Other types of data –ranks, e.g. preference between treatments –percentages, e.g. % oxygen uptake –rates or ratios, e.g. numbers of infant deaths/1000 –scores, e.g. Apgar score for evaluating new-born babies –visual analogue scales, e.g. perception of pain –survival data – two components, outcome and time to outcome

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Descriptive Statistics For continuous variables there are a number of useful descriptive statistics –Mean - equal to the sum of the observations divided by the number of observations, also known as the arithmetic mean –Median - the value that comes half-way when the data are ranked in order –Mode - the most common value observed –Standard Deviation - is a measure of the average deviation (or distance) of the observations from the mean –Standard Error of the mean - is measure of the uncertainty of a single sample mean as an estimate of the population mean

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Data Distributions Frequency distribution –If there are more than about 20 observations, a useful first step in summarizing quantitative data is to form a frequency distribution. This is a table showing the number of observations at different values or within certain ranges. If this is then plotted as a bar diagram a frequency distribution is obtained.

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The Normal Distribution In practice it is found that a reasonable description of many variables is provided by the normal distribution (Gaussian distribution). The curve of the normal distribution is symmetrical about the mean and bell-shaped. The bell is tall and narrow for small standard deviations, and short and wide for large ones.

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Descriptives 3.18139.515E-02 2.9946 3.3679 2.5920 1.8000 15.174 3.8954.10 33.40 33.30 2.3000 3.115.060 12.507.119 Mean Lower Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis DOD StatisticStd. Error

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Comparative statistics When there are two or more sets of observations from a study there are two types of design that must be distinguished: independent or paired. The design will determine the method of statistical analysis If the observations are from different groups of individuals, e.g. ages of males and females, or spectacle use in diabetics/non-diabetics, then the data is independent. The sample size may vary from group to group

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If each set of observations is made on the same group of individuals, e.g. WBC count pre- and post- treatment, then the data is said to be paired. This indicates that the observations are on the same individuals rather than from independent samples, and so we have the same number of observations in each set of data

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Independent data With independent continuous data, we are interested in the mean difference between the groups, but the variability between subjects becomes important. This is because the two sample t test (the most common test used), is based on the assumption that each set of observations is sampled from a population with a Normal Distribution, and that the variances of the two populations are the same.

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Non-parametric test If the continuous data is not normally distributed, or the standard deviations are very different, a non- parametric alternative to the t test known as the Mann-Whitney test can be utilised (another derivation of the same test is due to Wilcoxon)

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T-test

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Mann-Whitney Test

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Neutrophil engraftment following allogeneic SCT for CML 14188.1%1911.9%160100.0% 12295.3%64.7%128100.0% TBIDOSE 12 14.4 NEUTS NPercentN N ValidMissingTotal Cases

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Descriptives 22.9787.5816 21.8289 24.1286 22.5816 22.0000 47.692 6.9060 11.00 56.00 45.00 9.0000 1.162.204 3.184.406 26.6148.5544 25.5172 27.7123 26.1184 26.0000 37.495 6.1233 15.00 53.00 38.00 7.2500 1.453.219 4.157.435 Mean Lower Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean Lower Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis TBIDOSE 12 14.4 NEUTS StatisticStd. Error

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Descriptives 32.78911.6694 29.4857 36.0924 30.5556 29.5000 356.703 18.8866 14.00 186.00 172.00 11.7500 5.244.214 37.479.425 42.87763.9481 35.0417 50.7134 37.1973 27.0000 1527.572 39.0842 14.00 185.00 171.00 18.0000 2.368.244 4.780.483 Mean Lower Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean Lower Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis TBIDOSE 12 14.4 PLATES StatisticStd. Error

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Test Statistics 6172.5005543.500 11023.50015554.500 -.204-4.977 0.830.0006 Mann-Whitney U Wilcoxon W Z P-value (2-tailed) PLATESNEUTS

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Describing continuous data If the data is normally distributed –Mean and standard deviation If the data is skewed or non-normally distributed or is from a small sample (N<20) –Median and range

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Comparison of several means Data sets comprising more than two groups are common, and their analysis often involves the comparison of the means for the component subgroups. It is obviously possible to compare each pair of groups using t tests, but this is not a good approach. It is far better to use a single analysis that enables us to look at all the data in one go, and the method of choice is called analysis of variance If the data are not normally distributed or have different variances, a non-parametric equivalent to the analysis of variance can be used, and is known as the Kruskal-Wallis test

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Paired data When we have more than one group of observations it is vital to distinguish the case where the data are paired from that where the groups are independent. Paired data arise when the same individuals are studied more than once, usually in different circumstances. Also, when we have two different groups of subjects who have been individually matched, for example in a matched pair case-control study, then we should treat the data as paired.

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A one sample t test is used to examine the data. The value t is calculated from – t = sample mean - hypothesised mean standard error of sample mean In a paired analysis where one set of observations are subtracted from the other set, the hypothesised mean is zero. Thus the calculation of the t statistic reduces to –t = sample mean / standard error of sample mean The non-parametric equivalent to this test is the Wilcoxon matched pairs signed rank sum test

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Wilcoxon Signed Ranks Test

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Telomere length in Dyskeratosis Congenita

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Comparison of groups : continuous data Paired on non-paired? If non-paired and normally distributed with similar variances : T-test If non-paired non-normally distributed or with non-similar variances or very small numbers : Mann-Whitney test Paired data – paired t-test or Wilcoxon Signed Ranks Test

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Comparing Proportions Qualitative or categorical data is best presented in the form of table, such that one variable defines the rows, and the categories for the other variable define the columns. Thus in a European study of ASCT for HD, patient gender was compared between the UK and Europe The data are arranged in a contingency table Individuals are assigned to the appropriate cell of the contingency table according to their values for the two variables

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COUNTRYG * PSEX Crosstabulation Count 166108281454 100160260 167109881714 europe uk COUNTRYG Total FemaleMale PSEX Total

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COUNTRYG * PSEXG Crosstabulation 8286101438 57.6%42.4%100.0% 83.8%85.9%84.7% 160100260 61.5%38.5%100.0% 16.2%14.1%15.3% 9887101698 58.2%41.8%100.0% Count % within COUNTRYG % within PSEXG Count % within COUNTRYG % within PSEXG Count % within COUNTRYG % within PSEXG europe uk COUNTRYG Total 1.002.00 PSEXG Total

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Chi-squared test ( 2 ) A chi-squared test ( 2 ) is used to test whether there is an association between the row variable and the column variable. When the table has only two rows or two columns this is equivalent to the comparison of proportions.

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The first step in interpreting contingency table data is to calculate appropriate proportions or percentages. The chi-squared test compares the observed numbers in each of the four categories and compares them with the numbers expected if there were no difference between the distribution of patient gender The greater the differences between the observed and expected numbers, the larger the value of 2 and the less likely it is that the difference is due to chance

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COUNTRYG * PSEXG Crosstabulation 8286101438 57.6%42.4%100.0% 83.8%85.9%84.7% 160100260 61.5%38.5%100.0% 16.2%14.1%15.3% 9887101698 58.2%41.8%100.0% Count % within COUNTRYG % within PSEXG Count % within COUNTRYG % within PSEXG Count % within COUNTRYG % within PSEXG europe uk COUNTRYG Total 1.002.00 PSEXG Total

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Chi-Square Tests 1.418 b 1.234 1.2601.262 1.4281.232.246.131 1.4171.234 1698 Pearson Chi-Square Continuity Correction a Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases Valuedf Asymp. Sig. (2-sided) Exact Sig. (2-sided) Exact Sig. (1-sided) Computed only for a 2x2 table a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 108.72. b.

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Fishers Exact Test When the overall total of the table is less than 20, or if it is between 20 and 40 with the smallest of the four expected values is less than 5, then Fishers Exact Test should be used.

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Crosstab 15 100.0% 88.2%83.3% 213 66.7%33.3%100.0% 11.8%100.0%16.7% 17118 94.4%5.6%100.0% Count % within SURV % within TRMV Count % within SURV % within TRMV Count % within SURV % within TRMV.00 1.00 SURV Total DISG 3.00.001.00 TRMV Total

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Chi-Square Tests 5.294 b 1.021.8471.357 3.9051.048.167 5.0001.025 18 Pearson Chi-Square Continuity Correction a Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases DISG 3.00 Valuedf Asymp. Sig. (2-sided) Exact Sig. (2-sided) Exact Sig. (1-sided) Computed only for a 2x2 table a. 3 cells (75.0%) have expected count less than 5. The minimum expected count is.17. b.

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The chi-squared test can also be applied to larger tables, generally called r x c tables, where r denotes the number of rows in the table, and c the number of columns. The standard chi-squared test for a 2 x c table is a general test to assess whether there are differences among the c proportions. When the categories in the columns have a natural order, however, a more sensitive test is to look for an increasing (or decreasing) trend in the proportions over the columns. This trend can be tested using the chi-squared test for trend.

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Cesarean section Shoe Size <444.555.56Total Yes576781043 No1728364146140308 In the table below the relation between frequency of Cesarean section and maternal foot size is presented

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The standard chi-squared test of this 2 x 6 table gives and a 2 value of 9.29, with 5 d.f., for which P=0.098. Analysis of the data for trend gives a 2 trend = 8.02, with 1 d.f. (P=0.005). Thus there is strong evidence of a linear trend in the proportion of women giving birth by Cesarean section in relation to shoe size. This relation is not causal, but reflects that shoe size is a convenient indicator of small pelvic size

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Categorical data – comparing proportions Studies where there are 2 groups and the total number of patients > 40 : Chi-squared test Studies where there are 2 groups and the total number of patients < 40 or if more than 40 and a single cell has less than 5 : Fishers Exact Test Studies where there are more than 2 groups but not ordered : - Chi-squared test Studies where there are more than 2 groups which are ordered : - Chi-squared test for trend

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Exploring the relationship between two variables Three possible purposes : –a.) assess association e.g. body weight and blood pressure –b.) prediction e.g. height and weight –c.) assess agreement e.g. blood pressure measurement

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Correlation Method for investigating the linear association between two continuous variables The association is measured by the correlation coefficient A correlation between two variables shows that they are associated but does not necessarily imply a cause and effect relationship

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A t test is used to test whether the correlation coefficient obtained is significantly different from zero, or in other words whether the observed correlation could simply be due to chance The significance level is a function of both the size of the correlation coefficient and the number of observations. A weak correlation may therefore be statistically significant if based on a large number of observations, while a strong correlation may fail to achieve significance if there are only a few observations

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P=0.015

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P=<0.0001

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Problems with correlation analyses Biological systems are multifactoral so a simple two-way correlation may not be a true reflection of what is being observed Spurious correlations

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Assessing agreement Neither correlation nor linear regression are appropriate There may be a very high correlation, but one method gives a systematically higher/lower reading Linear regression, the data is not independent The only appropriate way is to subtract one observation from the other, and plot against an index variable

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Correlation between PCR and TAQman for measuring MRD 1.000 107.7391.000 0.0006 107 Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N PCR TAQ PCRTAQ

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Paired Samples Test 2.830E-028.117E-027.847E-031.274E-024.386E-023.606106 0.0002PCR - TAQ MeanStd. Deviation Std. Error MeanLowerUpper 95% Confidence Interval of the Difference Paired Differences tdfSig. (2-tailed)

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Linear regression Linear regression gives the equation of the straight line that describes how the y variable increases (or decreases) with an increase in the x variable. y is commonly called the dependent variable, and x the independent, or explanatory variable A t test is used to test whether the gradient b differs significantly from a specified value (usually zero)

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Assumptions – for any value of x, y must be normally distributed – the magnitude of the scatter of the points about the regression line is the same throughout the length of the line –the relation between the two variables should be linear

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Practical application Y = mx + c Telomere length = age * -0.049 + 17.89 Substituting in the above equation for ages of 30 and 60 16.42 = 30*-0.049 +17.89 14.95 = 60*-0.049 +17.89

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Survival data Has 2 components The event of interest and the time to the event Special statistical methods are required – it is not appropriate to use tests for categorical data

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Life Table Analysis Survival data are usually summarised as survival or Kaplan-Meier curves Based on a series of conditional probabilities For example, the probability of a patient surviving 10 days after a transplant, is the probability of surviving nine days, multiplied by the probability of surviving the 10 th day given that the patient survived the first nine days.

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04080120160200240280320 Days post BMT 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P a t i e n t n u m b e r Alive Dead

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Table 1. Life table for fifteen patients who received an allogeneic stem cell transplant Time (days)StatusNumber at riskProbability of survival Standard error 16*0151.00 261140.930.069 661130.860.094 69*012 741110.780.113 82*010 88190.690.129 89*08 117*07 133*06 144*05 172*04 252*03 291*02 305*01

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050100150200250300350 0 20 40 60 80 100 Days post BMT Probability %

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Outcomes suitable for Kaplan- Meier analyses Survival (event of interest is death, patients alive are censored) Disease-free survival (events of interest are either death or disease relapse, patients alive and in remission are censored) Primary graft failure Acute graft versus host disease

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012345 Years post BMT 0 20 40 60 80 100 P r o b a b i l i t y ( % ) Overall and leukaemia-free survival for 111 patients with CML 67% LFS OS 45% in CP allografted with stem cells from HLA-identical sibling donors HH/ICSM May 2003

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060120180240300360 Days post BMT 0 5 10 15 20 P r o b a b i l i t y o f g r a f t f a i l u r e ( % ) Graft failure following BMT for 1stCP CML with a VUD ICSM/HH April 13.2 Gy (n=57) 9% 14.4 Gy (n=44) 0%

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0285684112140168196224252280308336364 Days post BMT 0 20 40 60 80 100 P r o b a b i l i t y o f C M V r e a c t i v a t i o n ( % ) CMV reactivation following BMT with a VUD ICSM/HH May 35% 43% effect of ganciclovir treatment prophylactic treatment (n=49) post infection treatment (n=72)

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Use of computers for data collection/analysis Decide what data needs collecting (for statistical purposes) and then try if appropriate design a form (this is best done in a database, eg Microsoft ACCESS) Get the computer to do as much of the work as possible. ie calculation of ages, surface area etc Think ahead to what format the spreadsheet/stats package requires the data to be in

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For analysis purposes, its much easier to work with numbers and codes, as opposed to descriptions ie instead of male/female or m/f, use 1 or 2 Use a code to identify missing data, eg 999 or something unlikely Check the data before analysis, get descriptive statistics Use appropriate statistical methods Statistical packages - SPSS, BMDP, STATA, Statgraphics, MINITAB, STATXACT, GENSTAT, SAS

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Presentation of results Where possible give actual P values rather than ranges –ie P=0.041 rather than P<0.05 If a P value is not significant give the actual value and not just NS –ie P=0.15 rather than P=NS When presenting data it may be more useful to present confidence intervals rather than a P-value –ie lens A was more durable than lens B by 2.4 days (P=0.03), it might be more informative to write - lens A was more durable than lens B by 2.4 days (95%CI 0.3-4.5days)

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It is not necessary to give test results – ie t=33.5, 28 dof, P=0.0001 If a continuous variable is normally distributed present, as a description of the data, the mean and standard deviation, if not normally distributed, a median and range Dont quote more significant figures than necessary – ie mean patient age 34.2550 (std dev 11.4337), 34.3 (std dev 11.4) will suffice

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Writing the statistics section in a paper If power calculations were used to calculate the sample sizes, details should be given – eg Based on samples sizes of x in each arm, we should have been able to detect a difference of y given 80% power at a significance level of 0.05. State which statistical tests were used (reference obscure ones). –eg in order to investigate the differences between the groups, a t-test was used for continuous data, and a chi- squared test for categorical data

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If applicable, state whether standard deviations or standard errors are quoted State whether p-values are from one or two- tailed tests – eg all quoted p-values are two-tailed Not necessary to quote which stats package was used

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Suggested Reading Material Essentials of Medical Statistics –Betty Kirkwood Practical Statistics for Medical Research – Doug Altman Statistical Methods in Medical Research – Armitage and Berry

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Summary If at all possible - consult a statistician before starting your study Get a feel of your data by plotting results - dont rely on descriptive statistics alone Use appropriate statistical tests, not those that give the best results

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