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Unit 1 Overview Significance – How strong is the evidence of an effect? (Chapter 1) Estimation – How large is the effect? (Chapter 2) Generalization – How broadly do the conclusions apply? (Chapter 3) Causation – Can we say what caused the observed difference? (Chapter 4) Chapter 1 Significance: How Strong is the Evidence? Section 1.1: Introduction to Chance Models Organ Donation Study 78.6% in neutral group agreed 41.8% in the opt-in group agreed The researchers found these results to be statistically significant. This means that if the recruitment method made no difference in the proportion that would agree, results as different as we found would be unlikely to arise by random chance. Dolphin Communication Can dolphins communicate abstract ideas? In an experiment done in the 1960s, Doris was instructed which of two buttons to push. She then had to communicate this to Buzz (who could not see Doris). If he picked the correct button, both dolphins would get a reward. What are the observational units and variables in this study? Dolphin Communication In one set of trials, Buzz chose the correct button 15 out of 16 times. Based on these results, do you think Buzz knew which button to push or is he just guessing? How might we justify an answer? How might we model this situation? Modeling Buzz and Doris Flip Coins One Proportion Applet Simulation vs. Real Study coin flip = guess by Buzz heads = correct guess tails = wrong guess chance of probability of correct button heads = ½ = when Buzz is just guessing one set of 16 coin flips = one set of 16 attempts by Buzz Three S Strategy Statistic: Compute the statistic from the observed data. Simulate: Identify a model that represents a chance explanation. Repeatedly simulate values of the statistic that could have happened when the chance model is true and form a distribution. Strength of evidence: Consider whether the value of the observed statistic is unlikely to occur when the chance model is true. Buzz and Doris Redo Instead of a canvas curtain, Dr. Bastian constructed a wooden barrier between Buzz and Doris. When tested, Buzz pushed the correct button only 16 out of 28 times. Are these results statistically significant? Let’s go to the applet to check this out. Exploration 1.1: Can Dogs Understand Human Cues? (pg. 1-12) Dogs were positioned 2.5 m from experimenter. On each side of the experimenter were two cups. The experimenter would perform some human cue (pointing, bowing or looking) towards one of the cups. (Non-human cues were also done.) We will look at Harley’s results. Section 1.2: Measuring Strength of Evidence In the previous section we preformed tests of significance. In this section we will make things slightly more complicated, formalize the process, and define new terminology. We could take a look at Rock-Paper-Scissors-Lizard-Spock Scissors cut paper Paper covers rock Rock crushes lizard Lizard poisons Spock Spock smashes scissors Scissors decapitate lizard Lizard eats paper Paper disproves Spock Spock vaporizes rock (and as it always has) Rock crushes scissors RPS Rock-Paper-Scissors Rock smashes scissors Paper covers rock Scissors cut paper Are these choices used in equal proportions (1/3 each)? One study suggests that scissors are chosen less than 1/3 of the time. Rock-Paper-Scissors Suppose we are going to test this with 12 players each playing once against a computer. What are the observational units? What is the variable? Even though there are three outcomes, we are focusing on whether the player chooses scissors or not. This is called a binary variable since we are focusing on 2 outcomes (not both necessarily equally likely). Terminology: Hypotheses When conducting a test of significance, one of the first things we do is give the null and alternative hypotheses. The null hypothesis is the chance explanation. Typically the alternative hypothesis is what the researchers think is true. Hypotheses from Buzz and Doris Null Hypothesis: Buzz will randomly pick a button. (He chooses the correct button 50% of the time, in the long run.) Alternative Hypothesis: Buzz understands what Doris is communicating to him. (He chooses the correct button more than 50% of the time, in the long run.) These hypotheses represent the parameter (long run behavior) not the statistic (the observed results). Hypotheses for R-P-S in words Null Hypothesis: People playing RockPaper-Scissors will equally choose between the three options. (In particular, they will choose scissors one-third of the time, in the long run.) Alternative Hypothesis: People playing Rock-Paper-Scissors will choose scissors less than one-third of the time, in the long run. Note the differences (and similarities) between these hypotheses and those for Buzz and Doris. Hypotheses for R-P-S using symbols H0: π = 1/3 Ha: π < 1/3 where π is players’ true probability of throwing scissors Setting up a Chance Model Because the Buzz and Doris example had a 50% chance outcome, we could use a coin to model the outcome from one trial. What could we do in the case of Rock-PaperScissors? Same Three S Strategy as Before Statistic: Compute the statistic from the observed data. [In a class of 12 students, 2 picked scissors. This sample proportion can be described using the symbol (p-hat)]. Simulate: Identify a model that represents a chance explanation. Repeatedly simulate values of the statistic that could have happened when the chance model is true and form a distribution. Strength of evidence: Consider whether the value of the observed statistic is unlikely to occur when the chance model is true. Applet We will use the One Proportion Applet for our test. This is the same applet we used last time except now we will change the proportion under the null hypothesis. Let’s go to the applet and run the test. (Notice the use of symbols in the applet.) Null Distribution P-value The p-value is the proportion of the simulated statistics in the null distribution that are at least as extreme (in the direction of the alternative hypothesis) as the value of the statistic actually observed in the research study. We should have seen something similar to this in the applet Proportion of samples: 938/5000 = 0.1876 What can we conclude? Do we have strong evidence that less than 1/3 of the time scissors gets thrown? How small of a p-value would you say gives strong evidence? Remember the smaller the p-value, the stronger the evidence against the null. Guidelines for evaluating strength of evidence from p-values p-value >0.10, not much evidence against null hypothesis 0.05 < p-value < 0.10, moderate evidence against the null hypothesis 0.01 < p-value < 0.05, strong evidence against the null hypothesis p-value < 0.01, very strong evidence against the null hypothesis What can we conclude? So we do not have strong evidence that fewer than 1/3 of the time scissors is thrown. Does this mean we can conclude that 1/3 of the time scissors is thrown? Is it plausible that 1/3 of the time scissors is thrown? Are other values plausible? Which ones? What could we do to have a better chance of getting strong evidence for our alternative hypothesis? Summary The null hypothesis (H0) is the chance explanation. (=) The alternative hypothesis (Ha) is you are trying to show is true. (< or >) A null distribution is the distribution of simulated statistics that represent the chance outcome. The p-value is the proportion of the simulated statistics in the null distribution that are at least as extreme as the value of the observed statistic. Summary The smaller the p-value, the stronger the evidence against the null. P-values less than 0.05 provide strong evidence against the null. π is the population parameter is the sample proportion Exploration 1.2 (pg 1-25) Can people tell the difference between bottled and tap water? Alternative Measure of Strength of Evidence Section 1.3 Criminal Justice System vs. Significance Tests Innocent until proven guilty. We assume a defendant is innocent and the prosecution has to collect evidence to try to prove the defendant is guilty. Likewise, we assume our chance model (or null hypothesis) is true and we collect data and calculate a sample proportion. We then show how unlikely our proportion is if the chance model is true. Criminal Justice System vs. Significance Tests If the prosecution shows lots of evidence that goes against this assumption of innocence (DNA, witnesses, motive, contradictory story, etc.) then the jury might conclude that the innocence assumption is wrong. If after we collect data and find that the likelihood (p-value) of such a proportion is so small that it would rarely occur by chance if the null hypothesis is true, then we conclude our chance model is wrong. Review In the water tasting exploration, you could have obtained a null distribution similar to the one shown here. (H0: π = 0.25, Ha: π < 0.25 and = 3/27 = 0.1111) • • • • What What What What does a single dot represent? does the whole distribution represent? is the p-value for this simulation? does this p-value mean? More Review The null hypothesis is the chance explanation. Typically the alternative hypothesis is what the researchers think is true. The p-value is the proportion of outcomes in the null distribution that are at least as extreme as the value of the statistic actually observed in the study. Small p-values are evidence against the null. Strength of Evidence P-values are one measure for the strength of evidence and they are, by far, the most frequently used. P-values essentially are measures of how far the sample statistic is away from the parameter under the null hypothesis. Another measure for this distance we will look at today is called the standardized statistic. Heart Transplant Operations Example 1.3 Heart Transplants The British Medical Journal (2004) reported that heart transplants at St. George’s Hospital in London had been suspended after a spike in the mortality rate Of the last 10 heart transplants, 80% had resulted in deaths within 30 days This mortality rate was over five times the national average. The researchers used 15% as a reasonable value for comparison. Heart Transplants Does a heart transplant patient at St. George’s have a higher probability of dying than the national rate of 0.15? Observational units Variable The last 10 heart transplantations If the patient died or not Parameter The actual long-run probability of a death after a heart transplant operation at St. George’s Heart Transplants Null hypothesis: Death rate at St. George’s is the same as the national rate (0.15). Alternative hypothesis: Death rate at St. George’s is higher than the national rate. H0: = 0.15 Ha: > 0.15 Our statistic is 8 out of 10 or 0.80 Heart Transplants Simulation Null distribution of 1000 repetitions of drawing samples of 10 “patients” where the probability of death is equal to 0.15. What is the pvalue? Heart Transplants Strength of Evidence Our p-value is 0, so we have very strong evidence against the null hypothesis. Even with this strong evidence, it would be nice to have more data. Researchers examined the previous 361 heart transplantations at St. George’s and found that 71 died within 30 days. Our new statistic is 71/361 ≈ 0.197 Heart Transplants Here is a null distribution and p-value based on the new statistic. Heart Transplants The p-value was about 0.007 We still have very strong evidence against the null hypothesis, but not quite as strong as the first case Another way to measure strength of evidence is to standardize the observed statistic The Standardized Statistic The standardized statistic is the number of standard deviations our sample statistic is above the mean of the null distribution. statistic − mean of null distribution z = standard deviation of null distribution For a single proportion, we will use the symbol z for standardized statistic. The standardized statistic Here are the standardized statistics for our two studies. 0.80 − 0.15 = = 5.70 0.114 0.197 − 0.15 = = 2.47 0.019 In the first, our observed statistic was 5.70 standard deviations above the mean. In the second, our observed statistic was 2.47 standard deviations above the mean. Both of these are very strong, but we have stronger evidence against the null in the first. Guidelines for strength of evidence If a standardized statistic is below -2 or above 2, we have strong evidence against the null. Standardized Statistic Evidence Against Null between -1.5 and 1.5 not much below -1.5 or above 1.5 moderate below -2 or above 2 strong below -3 or above 3 very strong Which is Bob and which is Tim? Do People Use Facial Prototyping? Exploration 1.3 Impacting Strength of Evidence Section 1.4 Introduction We’ve now looked at tests of significance and have seen how p-values and standardized statistics give information about the strength of evidence against the null hypothesis. Today we’ll explore factors that affect strength of evidence. Bob or Tim? When the statistic is farther away from the proportion in the null, there is stronger evidence against the null. = 0.82 = 0.65 Predicting Elections from Faces Example 1.4 Predicting Elections Do voters make judgments about candidates based on facial appearances? More specifically, can you predict an election by choosing the candidate whose face is more competent-looking? Participants were shown two candidates and asked who has the more competentlooking face. Who has the more competent looking face? 2004 Senate Candidates from Wisconsin Winner Loser Bonus: One is named Tim and the other is Russ. Which name is the one on the left? 2004 Senate Candidates from Wisconsin Russ Tim Predicting Elections They determined which face was the more competent for the 32 Senate races in 2004. What are the observational units? The 32 Senate races What is the variable measured? If the method predicted the winner correctly Predicting Elections Null hypothesis: The probability this method predicts the winner equals 0.5. (H0: = 0.5) Alternative hypothesis: The probability this method predicts the winner is greater than 0.5. (Ha∶ > 0.5) This method predicted 23 of 32 races, hence = 23/32 ≈ 0.719, or 71.9%. Predicting Elections 1000 simulated sets of 32 races using the One Proportion applet. Predicting Elections With a p-value of 0.009 we have strong evidence against the null hypothesis. When we calculate the standardized statistic we again show strong evidence against the null. 0.7188 − 0.501 = = 2.42. 0.09 What do the p-value and standardized statistic mean? What effects the strength of evidence? 1. 2. 3. The difference between the observed statistic () and null hypothesis parameter (). Sample size. If we do a one or two-sided test. Difference between and What if researchers predicted 26 elections instead of 23? 26/32 = 0.8125 never occurs just by chance hence the p-value is 0. Difference between and The farther away the observed statistic is from the average value of the null distribution (or ), the more evidence there is against the null hypothesis. Sample Size Suppose the sample proportion stays the same, do you think increasing sample size will increase, decrease, or have no impact on the strength of evidence against the null hypothesis? Sample Size The null distribution changes as we increase the sample size from 32 senate races to 128 races to 256 races. As the sample sizes increases, the variability (standard deviation) decreases. Sample Size What does decreasing variability mean for statistical significance (with same sample proportion)? 32 elections 128 elections p-value = 0.009 and z = 2.42 p-value = 0 and z =5.07 256 elections Even stronger evidence p-value = 0 and z = 9.52 Sample Size As the sample size increases, the variability decreases. Therefore, as the sample size increases, the evidence against the null hypothesis increases (as long as the sample proportion stays the same and is in the direction of the alternative hypothesis). Two-Sided Tests What if researchers were wrong; instead of the person with the more competent face being elected more frequently, it was actually the less frequently? H0: = 0.5 Ha: > 0.5 With this alternative, if we go a sample proportion less than 0.5, we would get a very large p-value. This is a one-sided test. Often one-sided is too narrow In fact most research uses two-sided tests. Two-Sided Tests In a two-sided test the alternative can be concluded when sample proportions are in either tail of the null distribution. Null hypothesis: The probability this method predicts the winner equals 0.50. (H0: π = 0.50) Alternative hypothesis: The probability this method predicts the winner is not 0.50. (Ha: π ≠ 0.50) Two-Sided Tests The change to the alternative hypothesis also effects how we compute the p-value. Remember that the p-value is the probability (assuming the null hypothesis is true) of obtaining a proportion that is equal to or more extreme than the observed statistic In a two-sided test, more extreme goes in both directions. Two-Sided Tests Since our sample proportion was 0.7188 and 0.7188 is 0.2188 above 0.5, we also need to look at 0.2188 below 0.5. (This gets a bit more complicated when the distribution is not symmetric, but the applet will do all the work for you.) Hence the p-value will include all simulated proportions 0.7188 and above as well as those 0.2812 and below. Two-Sided Tests 0.7188 or greater was obtained 9 times 0.2812 or less was obtained 8 times The p-value is (8 + 9 = 17)/1000 = 0.017. Two-sided tests increase the p-value (it about doubles) and hence decrease the strength of evidence. Two-sided tests are said to be more conservative. More evidence is needed to conclude alternative. Let’s check this out using our applet. Predicting House Elections Researchers also predicted the 279 races for the House of Representatives in 2004 The correctly predicted the winner in 189/279 ≈ 0.677, or 67.7% of the races. The House’s sample percentage (67.7%) is bit smaller than the Senate (71.9%), but that the sample size is larger (279) than for the senate races (32). Do you expect the strength of evidence to be stronger, weaker, or essentially the same for the House compared to the Senate? Predicting House Elections Distance of the observed statistic to the null hypothesis value The statistic in the House is 0.677 compared to 0.719 in the Senate Slight decrease in the strength of evidence Sample size The sample size is almost 10 times as large (279 vs. 32) This will increase the strength of evidence Predicting House Elections Null distribution of 279 sample House races Simulated statistics ≥0.677 didn’t occur hence the p-value is 0 Predicting House Elections What about the standardized statistics? For the Senate it was 2.49 For the House is 5.90. The larger sample size for the House trumped its smaller proportion, so we have stronger evidence against the null using the data from the House. Competitive Advantage to Uniform Colors? Exploration 1.4 In four contact sports (boxing, tae kwon do, Greco–Roman wrestling and freestyle wrestling) in the 2004 Olympics, participants were randomly assigned a red or blue uniform. Researches then analyzed the results. Section 1.5 Normal Approximation (Theory-Based Test) Simulation-Based vs. Theory-Based We will now look at the more traditional method of determining a p-value through theory-based techniques. When we used simulation-based methods, we all got slightly different p-values. The more repetitions we would do, the closer our p-values will be to each other. In theory-based methods, we will use a theoretical distribution to model our null distribution and we will all get the same pvalue. Theory-Based Techniques Hopefully, you’ve noticed the shape of most of our simulated null distributions were quite predictable. We can predict this shape using normal distributions. When we do a test of significance using theory-based methods, only how our pvalues are found will change. Everything else will stay the same. The Null Distributions Our null distributions: Were typically bell shaped Centered at the proportion under the null Their width was dependent mostly on the sample size. The Normal Distribution Both of these are centered at 0.5. The one on the left represents samples of size 30. The one on the right represents samples of size 300. Both could be predicted using normal distributions. Examples from this chapter Which ones will normal distributions fit? When can I use a theory-based test that uses the normal distribution? The shape of the randomized null distribution is affected by the sample size and the proportion under which you are testing. The larger the sample size the better. The closer the null proportion is to 0.5 the better. A simple rule of thumb to follow is: You should have at least 10 successes and 10 failures in your sample to be fairly confident that a normal distribution will fit the simulated null distribution nicely. We will call guidelines like that above as validity conditions. Advantages and Disadvantages of Theory-Based Tests Advantages of theory-based tests No need to set up some randomization method Fast and Easy Can be done with a wide variety of software We all get the same p-value. Determining confidence intervals (we will do this next time) is much easier. Disadvantages of theory-based tests They will all come with some validity conditions (like the number of success and failures we have for a single proportion test). Example 1.5: Halloween Treats Researchers investigated whether children show a preference to toys or candy Test households in five Connecticut neighborhoods offered children two plates: One with candy One with small, inexpensive toys The researchers observed the selections of 283 trick-or-treaters between ages 3 and 14. Halloween Treats Null: The proportion of trick-or-treaters who choose candy is 0.5. Alternative: The proportion of trick-ortreaters who choose candy is not 0.5. H0: π= 0.5 Ha: π ≠ 0.5 Notice we are focusing on candy, but could have easily done this focusing on the toy. Halloween Treats 283 children were observed 148 (52.3%) chose candy 135 (47.7%) chose toys Let’s first run this test using oneproportion applet we have been using. When doing this notice what the shape, center and standard deviation of the null distribution. Predicting Standard Deviation Could you have predicted the center and shape of the null distribution? What about the standard deviation? This is a bit harder, but can easily done with the formula 1 − / where π is the proportion under the null and n is the sample size. 0.5(1−0.5) 283 = 0.0297. Theory-Based Inference These predictions work if we have a large enough sample size. We have 148 successes and 135 failures. Is the sample size large enough to use the theory-based method? Use the One Proportion applet to find the theory-based (normal approximation) p-value. Halloween Treats If half of the population of trick-ortreaters preferred candy, then there’s a 43.9% chance that a random sample of 283 trick-or-treaters would have 148 or more, or 135 or fewer, choose the candy. Since its not a small p-value, we don’t have strong (or even moderate) evidence that trick-or-treaters prefer one type of treat over the other. Standardized Statistic Notice that the standardized statistic in the applet is 0.77 (or sample proportion is 0.77 SD above the mean). Remember that a standardized statistic of more than 2 indicates that the sample result is far enough from the hypothesized value to be unlikely if the null were true. We had a standardized statistic that was not more than 2 (or even 1) so we don’t really have any evidence against the null. What happens when validity condition is not met? Suppose we’re testing 12 repetitions of the Rock-Paper-Scissors test and 1 of the 12 repetitions had scissors thrown. Use the One-Proportion applet to test the (we will use a less than alternative) and also look at the normal approximation results. H or T Exploration 1.5: Heads or Tails?