DOCSLIB.ORG

Understanding Statistics 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Understanding Statistics 3 RESEARCH/TOOLBOXTOOLBOX Understanding statistics 3 Elliot Abt Department of Dentistry, Illinois Masonic Medical Center, Chicago, Illinois, USA The last article in this series described the dif- Table 1 Type I and Type II errors ferent types and nature of data used in quan- Null Hypothesis (H ) is true Alternative Hypothesis (H ) is true titative research. This article will define and 0 1 examine the relationships between P-values, Fail to Reject Right decision Wrong decision Null Hypothesis Type II Error confidence intervals and sample size. False Negative Reject Wrong decision Right decision P-values Null Hypothesis Type I Error False Positive Central to the theme of hypothesis testing is the P-value, which is often misunderstood or misinterpreted. While it is often stated as have argued against this all or none phenome- variability in a sample, and generally does the probability of the data having arisen by non of statistical significance.2 At the very least, not change as the sample size increases chance, the P-value is actually the probabili- a study should report a specific P-value (i.e. n = sample size ty of obtaining the observed effect (or a more p=0.03) as this will provide a better estimate CI = confidence interval extreme value), given the null hypothesis is of the strength of the evidence against the null = mean or point estimate true.1 Traditionally, the level of significance hypothesis. Unfortunately, what P-values don’t is set at 5% (0.05), as at this level we are no tell us is the direction of the effect (is drug A This is the equation for a 95% CI which longer 'comfortable' that an observed effect better than B, or vice versa) and little about the corresponds to setting the level of statisti- is due to chance, but this is arbitrary, and it magnitude of the effect. For that, confidence cal significance at 5%. Other CIs can be is occasionally set at 1% or even 10%. intervals (CIs) are required. calculated for different levels of statisti- If P<0.05, this tells us that either two cal significance. These are basic equations groups are different, or perhaps that there is Intervals and sample size which change with increasing complexity of a 1 in 20 chance we are making an error. This Many are familiar with descriptive statistics; statistical tests. is referred to as a type I error, or a 'false posi- the presentation, organization, and tive', where we falsely reject the null hypoth- summarization of data such as simple P-values, confidence intervals, and esis. Conversely, if P>0.05, either there are percentages. Inferential statistics involve the null value no differences between groups, or we have extrapolating data from a sample to a A relationship exists between P-values, made a type II error, or falsely accepted the population, and CIs give readers a range confidence intervals and the null value. null hypothesis (table 1). This is often due to of values where we are confident the true If the 95% CI includes the null value, by a study which is underpowered, or lacking population lies.3 CIs are derived from the definition, then P>0.05. If the 95% CI does enough participants to show a difference standard error (SE), which, in turn is derived not include the null value, then P<0.05. It between groups if one actually exists. One from the sample size. That is, the larger the should be noted that the null value is not could question the ethics of conducting a study sample, the smaller the SE, and the always “0”. It varies depending on what type trial which does not have enough subjects to narrower the CI. This may be somewhat of data are used in a study. show differences (or associations) between intuitive, that the more participants, the The next article in this series will begin to groups. In studies with excessive numbers of more confident we are in the study results. explore basic statistical tests and give examples participants, very small differences between Thus, as sample size increases, CIs narrow, of dental studies using different data types. It groups (around 1%) are likely to be statisti- providing a more precise estimate of effect. will provide assistance with statistical analysis, cally significant, although not necessar- The relationship between sample size and CI s including interpretation of CIs and their effect ily clinically relevant. Thus, readers of the can be illustrated in the following two (basic) on clinical relevance. literature should look for a power (sample equations: size) calculation to estimate an appropriate SE = s ÷ √n 1. Altman DG. Practical Statistics for Medical Research. Section 8.5; Chapman and Hall/CRC. number of study participants. 95% CI = ± 1.96(SE) 2. Guyatt G, Jaeschke R, Heddle N, Cook D, Shannon Many practitioners have been trained to SE = standard error - which measures the H, and Walter S. Basic statistics for clinicians: 1. Hypothesis testing. CMAJ 1995;152: 27-32. see if the P-value is greater or less than 0.05. amount of variability in the sample mean 3. Kirkwood BR, and Sterne JAC. Essential Medical Statistics. 2nd Edition. Ch. 8; Blackwell Science Technically, P-values of 0.049 and 0.051 would and indicates how accurately the sample be on opposite sides of statistical significance, mean represents the mean of a population Evidence-Based Dentistry (2010) 11, 118. when, in fact, they are equivalent, and some s = standard deviation - describes the doi:10.1038/sj.ebd.6400761 118 © EBD 2010:11.4 © 2010 Macmillan Publishers Limited. All rights reserved.
Load more

Recommended publications
  • Confidence Intervals for the Population Mean Alternatives to the Student-T Confidence Interval
    Journal of Modern Applied Statistical Methods Volume 18 Issue 1 Article 15 4-6-2020 Robust Confidence Intervals for the Population Mean Alternatives to the Student-t Confidence Interval Moustafa Omar Ahmed Abu-Shawiesh The Hashemite University, Zarqa, Jordan, [email protected] Aamir Saghir Mirpur University of Science and Technology, Mirpur, Pakistan, [email protected] Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Abu-Shawiesh, M. O. A., & Saghir, A. (2019). Robust confidence intervals for the population mean alternatives to the Student-t confidence interval. Journal of Modern Applied Statistical Methods, 18(1), eP2721. doi: 10.22237/jmasm/1556669160 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Robust Confidence Intervals for the Population Mean Alternatives to the Student-t Confidence Interval Cover Page Footnote The authors are grateful to the Editor and anonymous three reviewers for their excellent and constructive comments/suggestions that greatly improved the presentation and quality of the article. This article was partially completed while the first author was on sabbatical leave (2014–2015) in Nizwa University, Sultanate of Oman. He is grateful to the Hashemite University for awarding him the sabbatical leave which gave him excellent research facilities. This regular article is available in Journal of Modern Applied Statistical Methods: https://digitalcommons.wayne.edu/ jmasm/vol18/iss1/15 Journal of Modern Applied Statistical Methods May 2019, Vol.
    [Show full text]
  • Projections of Education Statistics to 2022 Forty-First Edition
    Projections of Education Statistics to 2022 Forty-first Edition 20192019 20212021 20182018 20202020 20222022 NCES 2014-051 U.S. DEPARTMENT OF EDUCATION Projections of Education Statistics to 2022 Forty-first Edition FEBRUARY 2014 William J. Hussar National Center for Education Statistics Tabitha M. Bailey IHS Global Insight NCES 2014-051 U.S. DEPARTMENT OF EDUCATION U.S. Department of Education Arne Duncan Secretary Institute of Education Sciences John Q. Easton Director National Center for Education Statistics John Q. Easton Acting Commissioner The National Center for Education Statistics (NCES) is the primary federal entity for collecting, analyzing, and reporting data related to education in the United States and other nations. It fulfills a congressional mandate to collect, collate, analyze, and report full and complete statistics on the condition of education in the United States; conduct and publish reports and specialized analyses of the meaning and significance of such statistics; assist state and local education agencies in improving their statistical systems; and review and report on education activities in foreign countries. NCES activities are designed to address high-priority education data needs; provide consistent, reliable, complete, and accurate indicators of education status and trends; and report timely, useful, and high-quality data to the U.S. Department of Education, the Congress, the states, other education policymakers, practitioners, data users, and the general public. Unless specifically noted, all information contained herein is in the public domain. We strive to make our products available in a variety of formats and in language that is appropriate to a variety of audiences. You, as our customer, are the best judge of our success in communicating information effectively.
    [Show full text]
  • Use of Statistical Tables
    TUTORIAL | SCOPE USE OF STATISTICAL TABLES Lucy Radford, Jenny V Freeman and Stephen J Walters introduce three important statistical distributions: the standard Normal, t and Chi-squared distributions PREVIOUS TUTORIALS HAVE LOOKED at hypothesis testing1 and basic statistical tests.2–4 As part of the process of statistical hypothesis testing, a test statistic is calculated and compared to a hypothesised critical value and this is used to obtain a P- value. This P-value is then used to decide whether the study results are statistically significant or not. It will explain how statistical tables are used to link test statistics to P-values. This tutorial introduces tables for three important statistical distributions (the TABLE 1. Extract from two-tailed standard Normal, t and Chi-squared standard Normal table. Values distributions) and explains how to use tabulated are P-values corresponding them with the help of some simple to particular cut-offs and are for z examples. values calculated to two decimal places. STANDARD NORMAL DISTRIBUTION TABLE 1 The Normal distribution is widely used in statistics and has been discussed in z 0.00 0.01 0.02 0.03 0.050.04 0.05 0.06 0.07 0.08 0.09 detail previously.5 As the mean of a Normally distributed variable can take 0.00 1.0000 0.9920 0.9840 0.9761 0.9681 0.9601 0.9522 0.9442 0.9362 0.9283 any value (−∞ to ∞) and the standard 0.10 0.9203 0.9124 0.9045 0.8966 0.8887 0.8808 0.8729 0.8650 0.8572 0.8493 deviation any positive value (0 to ∞), 0.20 0.8415 0.8337 0.8259 0.8181 0.8103 0.8206 0.7949 0.7872 0.7795 0.7718 there are an infinite number of possible 0.30 0.7642 0.7566 0.7490 0.7414 0.7339 0.7263 0.7188 0.7114 0.7039 0.6965 Normal distributions.
    [Show full text]
  • Startips …A Resource for Survey Researchers
    Subscribe Archives StarTips …a resource for survey researchers Share this article: How to Interpret Standard Deviation and Standard Error in Survey Research Standard Deviation and Standard Error are perhaps the two least understood statistics commonly shown in data tables. The following article is intended to explain their meaning and provide additional insight on how they are used in data analysis. Both statistics are typically shown with the mean of a variable, and in a sense, they both speak about the mean. They are often referred to as the "standard deviation of the mean" and the "standard error of the mean." However, they are not interchangeable and represent very different concepts. Standard Deviation Standard Deviation (often abbreviated as "Std Dev" or "SD") provides an indication of how far the individual responses to a question vary or "deviate" from the mean. SD tells the researcher how spread out the responses are -- are they concentrated around the mean, or scattered far & wide? Did all of your respondents rate your product in the middle of your scale, or did some love it and some hate it? Let's say you've asked respondents to rate your product on a series of attributes on a 5-point scale. The mean for a group of ten respondents (labeled 'A' through 'J' below) for "good value for the money" was 3.2 with a SD of 0.4 and the mean for "product reliability" was 3.4 with a SD of 2.1. At first glance (looking at the means only) it would seem that reliability was rated higher than value.
    [Show full text]
  • Appendix F.1 SAWG SPC Appendices
    Appendix F.1 - SAWG SPC Appendices 8-8-06 Page 1 of 36 Appendix 1: Control Charts for Variables Data – classical Shewhart control chart: When plate counts provide estimates of large levels of organisms, the estimated levels (cfu/ml or cfu/g) can be considered as variables data and the classical control chart procedures can be used. Here it is assumed that the probability of a non-detect is virtually zero. For these types of microbiological data, a log base 10 transformation is used to remove the correlation between means and variances that have been observed often for these types of data and to make the distribution of the output variable used for tracking the process more symmetric than the measured count data1. There are several control charts that may be used to control variables type data. Some of these charts are: the Xi and MR, (Individual and moving range) X and R, (Average and Range), CUSUM, (Cumulative Sum) and X and s, (Average and Standard Deviation). This example includes the Xi and MR charts. The Xi chart just involves plotting the individual results over time. The MR chart involves a slightly more complicated calculation involving taking the difference between the present sample result, Xi and the previous sample result. Xi-1. Thus, the points that are plotted are: MRi = Xi – Xi-1, for values of i = 2, …, n. These charts were chosen to be shown here because they are easy to construct and are common charts used to monitor processes for which control with respect to levels of microbiological organisms is desired.
    [Show full text]
  • STAT 22000 Lecture Slides Overview of Confidence Intervals
    STAT 22000 Lecture Slides Overview of Confidence Intervals Yibi Huang Department of Statistics University of Chicago Outline This set of slides covers section 4.2 in the text • Overview of Confidence Intervals 1 Confidence intervals • A plausible range of values for the population parameter is called a confidence interval. • Using only a sample statistic to estimate a parameter is like fishing in a murky lake with a spear, and using a confidence interval is like fishing with a net. We can throw a spear where we saw a fish but we will probably miss. If we toss a net in that area, we have a good chance of catching the fish. • If we report a point estimate, we probably won’t hit the exact population parameter. If we report a range of plausible values we have a good shot at capturing the parameter. 2 Photos by Mark Fischer (http://www.flickr.com/photos/fischerfotos/7439791462) and Chris Penny (http://www.flickr.com/photos/clearlydived/7029109617) on Flickr. Recall that CLT says, for large n, X ∼ N(µ, pσ ): For a normal n curve, 95% of its area is within 1.96 SDs from the center. That means, for 95% of the time, X will be within 1:96 pσ from µ. n 95% σ σ µ − 1.96 µ µ + 1.96 n n Alternatively, we can also say, for 95% of the time, µ will be within 1:96 pσ from X: n Hence, we call the interval ! σ σ σ X ± 1:96 p = X − 1:96 p ; X + 1:96 p n n n a 95% confidence interval for µ.
    [Show full text]
  • Problems with OLS Autocorrelation
    Problems with OLS Considering : Yi = α + βXi + ui we assume Eui = 0 2 = σ2 = σ2 E ui or var ui Euiuj = 0orcovui,uj = 0 We have seen that we have to make very specific assumptions about ui in order to get OLS estimates with the desirable properties. If these assumptions don’t hold than the OLS estimators are not necessarily BLU. We can respond to such problems by changing specification and/or changing the method of estimation. First we consider the problems that might occur and what they imply. In all of these we are basically looking at the residuals to see if they are random. ● 1. The errors are serially dependent ⇒ autocorrelation/serial correlation. 2. The error variances are not constant ⇒ heteroscedasticity 3. In multivariate analysis two or more of the independent variables are closely correlated ⇒ multicollinearity 4. The function is non-linear 5. There are problems of outliers or extreme values -but what are outliers? 6. There are problems of missing variables ⇒can lead to missing variable bias Of course these problems do not have to come separately, nor are they likely to ● Note that in terms of significance things may look OK and even the R2the regression may not look that bad. ● Really want to be able to identify a misleading regression that you may take seriously when you should not. ● The tests in Microfit cover many of the above concerns, but you should always plot the residuals and look at them. Autocorrelation This implies that taking the time series regression Yt = α + βXt + ut but in this case there is some relation between the error terms across observations.
    [Show full text]
  • On the Meaning and Use of Kurtosis
    Psychological Methods Copyright 1997 by the American Psychological Association, Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics standard deviation. The normal distribution has a kur- courses that distributions can be characterized in tosis of 3, and 132 - 3 is often used so that the refer- terms of central tendency, variability, and shape. With ence normal distribution has a kurtosis of zero (132 - respect to shape, virtually every textbook defines and 3 is sometimes denoted as Y2)- A sample counterpart illustrates skewness. On the other hand, another as- to 132 can be obtained by replacing the population pect of shape, which is kurtosis, is either not discussed moments with the sample moments, which gives or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- ~(X i -- S)4/n search articles, in spite of the fact that virtually every b2 (•(X i - ~')2/n)2' statistical package provides a measure of kurtosis.
    [Show full text]
  • The Probability Lifesaver: Order Statistics and the Median Theorem
    The Probability Lifesaver: Order Statistics and the Median Theorem Steven J. Miller December 30, 2015 Contents 1 Order Statistics and the Median Theorem 3 1.1 Definition of the Median 5 1.2 Order Statistics 10 1.3 Examples of Order Statistics 15 1.4 TheSampleDistributionoftheMedian 17 1.5 TechnicalboundsforproofofMedianTheorem 20 1.6 TheMedianofNormalRandomVariables 22 2 • Greetings again! In this supplemental chapter we develop the theory of order statistics in order to prove The Median Theorem. This is a beautiful result in its own, but also extremely important as a substitute for the Central Limit Theorem, and allows us to say non- trivial things when the CLT is unavailable. Chapter 1 Order Statistics and the Median Theorem The Central Limit Theorem is one of the gems of probability. It’s easy to use and its hypotheses are satisfied in a wealth of problems. Many courses build towards a proof of this beautiful and powerful result, as it truly is ‘central’ to the entire subject. Not to detract from the majesty of this wonderful result, however, what happens in those instances where it’s unavailable? For example, one of the key assumptions that must be met is that our random variables need to have finite higher moments, or at the very least a finite variance. What if we were to consider sums of Cauchy random variables? Is there anything we can say? This is not just a question of theoretical interest, of mathematicians generalizing for the sake of generalization. The following example from economics highlights why this chapter is more than just of theoretical interest.
    [Show full text]
  • The Bootstrap and Jackknife
    The Bootstrap and Jackknife Summer 2017 Summer Institutes 249 Bootstrap & Jackknife Motivation In scientific research • Interest often focuses upon the estimation of some unknown parameter, θ. The parameter θ can represent for example, mean weight of a certain strain of mice, heritability index, a genetic component of variation, a mutation rate, etc. • Two key questions need to be addressed: 1. How do we estimate θ ? 2. Given an estimator for θ , how do we estimate its precision/accuracy? • We assume Question 1 can be reasonably well specified by the researcher • Question 2, for our purposes, will be addressed via the estimation of the estimator’s standard error Summer 2017 Summer Institutes 250 What is a standard error? Suppose we want to estimate a parameter theta (eg. the mean/median/squared-log-mode) of a distribution • Our sample is random, so… • Any function of our sample is random, so... • Our estimate, theta-hat, is random! So... • If we collected a new sample, we’d get a new estimate. Same for another sample, and another... So • Our estimate has a distribution! It’s called a sampling distribution! The standard deviation of that distribution is the standard error Summer 2017 Summer Institutes 251 Bootstrap Motivation Challenges • Answering Question 2, even for relatively simple estimators (e.g., ratios and other non-linear functions of estimators) can be quite challenging • Solutions to most estimators are mathematically intractable or too complicated to develop (with or without advanced training in statistical inference) • However • Great strides in computing, particularly in the last 25 years, have made computational intensive calculations feasible.
    [Show full text]
  • Lecture 5 Significance Tests Criticisms of the NHST Publication Bias Research Planning
    Lecture 5 Significance tests Criticisms of the NHST Publication bias Research planning Theophanis Tsandilas !1 Calculating p The p value is the probability of obtaining a statistic as extreme or more extreme than the one observed if the null hypothesis was true. When data are sampled from a known distribution, an exact p can be calculated. If the distribution is unknown, it may be possible to estimate p. 2 Normal distributions If the sampling distribution of the statistic is normal, we will use the standard normal distribution z to derive the p value 3 Example An experiment studies the IQ scores of people lacking enough sleep. H0: μ = 100 and H1: μ < 100 (one-sided) or H0: μ = 100 and H1: μ = 100 (two-sided) 6 4 Example Results from a sample of 15 participants are as follows: 90, 91, 93, 100, 101, 88, 98, 100, 87, 83, 97, 105, 99, 91, 81 The mean IQ score of the above sample is M = 93.6. Is this value statistically significantly different than 100? 5 Creating the test statistic We assume that the population standard deviation is known and equal to SD = 15. Then, the standard error of the mean is: σ 15 σµˆ = = =3.88 pn p15 6 Creating the test statistic We assume that the population standard deviation is known and equal to SD = 15. Then, the standard error of the mean is: σ 15 σµˆ = = =3.88 pn p15 The test statistic tests the standardized difference between the observed mean µ ˆ = 93 . 6 and µ 0 = 100 µˆ µ 93.6 100 z = − 0 = − = 1.65 σµˆ 3.88 − The p value is the probability of getting a z statistic as or more extreme than this value (given that H0 is true) 7 Calculating the p value µˆ µ 93.6 100 z = − 0 = − = 1.65 σµˆ 3.88 − The p value is the probability of getting a z statistic as or more extreme than this value (given that H0 is true) 8 Calculating the p value To calculate the area in the distribution, we will work with the cumulative density probability function (cdf).
    [Show full text]
  • Statistics for Dummies Cheat Sheet from Statistics for Dummies, 2Nd Edition by Deborah Rumsey
    Statistics For Dummies Cheat Sheet From Statistics For Dummies, 2nd Edition by Deborah Rumsey Copyright © 2013 & Trademark by John Wiley & Sons, Inc. All rights reserved. Whether you’re studying for an exam or just want to make sense of data around you every day, knowing how and when to use data analysis techniques and formulas of statistics will help. Being able to make the connections between those statistical techniques and formulas is perhaps even more important. It builds confidence when attacking statistical problems and solidifies your strategies for completing statistical projects. Understanding Formulas for Common Statistics After data has been collected, the first step in analyzing it is to crunch out some descriptive statistics to get a feeling for the data. For example: Where is the center of the data located? How spread out is the data? How correlated are the data from two variables? The most common descriptive statistics are in the following table, along with their formulas and a short description of what each one measures. Statistically Figuring Sample Size When designing a study, the sample size is an important consideration because the larger the sample size, the more data you have, and the more precise your results will be (assuming high- quality data). If you know the level of precision you want (that is, your desired margin of error), you can calculate the sample size needed to achieve it. To find the sample size needed to estimate a population mean (µ), use the following formula: In this formula, MOE represents the desired margin of error (which you set ahead of time), and σ represents the population standard deviation.
    [Show full text]