Name: Objectives Lab 5 Statistics and Analysis After completing this introduction, students will be able to use the scientific process proficiently to answer scientific questions and construct new knowledge, including: Asking questions and defining hypotheses, Planning and carrying out investigations, Analyzing and interpreting data, Constructing explanations, Engaging in argument from data and, Communicating information. A. Inferential Statistics This is the one with math in it. Fortunately, statistics are so ubiquitous in the sciences that many tools are made to make it readily accessible and lightning quick. Last lab, we used descriptive statistics that attempt to describe the data collected. Today, we will be working with inferential statistics. Inferential statistics use data from a sample of the population to infer trends and generalizations from the entire population. This is most often done because collecting data on all members of a population may be realistically impossible. For example, our study is looking at a single animal species caught by camera traps in Gorongosa National Park. The individuals photographed do not represent all the members of that species, but we can use it to infer how all members of the species generally behave. As you might expect, the more we sample, the closer we get to sampling all members, thus our statistical analyses are more accurate. While there are many types of statistical tests out there, we will practice using the chi-square test. Before we move on, let's discuss a few similarities between most statistical tests. During your own project, your research may require you to run a different test (don't worry, your TA will be here to guide and help!), but interpreting results are (mostly) all the same. Statistical tests attempt to test a null hypothesis. Earlier, we used a hypothesis to explain some relation between an independent variable(s) and a dependent variable. The null hypothesis is the assumption there is no relationship between variables. For example, if you were hypothesizing that test scores are determined by amount of time studied, then the null hypothesis would be that the amount of time studied has no effect on test scores (i.e. someone who does not study will receive a grade equal to someone that studied a lot). Statistical tests often convey this test of the null hypothesis as a p value. You can think of the p value as the probability that differences in your data are due to chance. In science, we accept that if the p-value is less than 0.05 (p < 0.05), then we reject the null hypothesis. In other words, scientists accept a less-than 1 Name: 5% chance of data being different due to chance, as opposed to some mechanism or process. Some scientific studies may even reduce this p-value to 0.01 or 0.001 if they want to severely reduce making an error; this is commonly done in medicine to ensure medical practices are safe for the vast public. STOP: Answer #1-2 in your Lab 5 Report to summarize your understanding of inferential statistics. Then continue on to Section B to conduct a statistical test on the data we collected in Lab 4. B. Stats Tool #1: Chi-Square (x) A chi-square statistical test is useful when we have two independent variables in a contingency table. In our case, this would be 'season' and 'habitat.' A chi-square test will determine if there is a relationship between our two variables by looking at our observed data (the data in our contingency table from last week) and the expected data that we would see if there were no relationship at all. 1. Start by opening your spreadsheet from Lab 4. 2. To calculate the chi-square statistical test, we are going to use a chi-square calculator available at: http://www.vassarstats.net/newcs.html 3. When you open the calculator, the first thing that you must do is enter the number of rows and columns in your contingency table, as shown below. Since we have 4 rows (seasons) and 3 columns (habitats), make sure your selection screen looks as follows: Select the number of rows: 2 3 4 5 4 Select the number of columns: 2 3 4 5 3 Data Entry B B B3 B4 B5 Totals A1 A2 ----- ----- A3 A4 A5 Totals ----- ----- Reset Calculate 4. Then, enter your data into the contingency table outline in yellow. You do not need to enter totals; they will be calculated for you. Then click "Calculate." 5. The calculator will give you a p-value. Remember, this is the probability that the differences in your data occur from random chance. The image on the next page shows my calculations for porcupines in Gorongosa National Park. Notice the p-value obtained is 0.001 which is less than 0.05, thus I reject the null hypothesis: these porcupines are switching their habitat use based on season! 2 Name: Data Entry B1 B B3 B4 B5 Totals A1 108 8 153 ----- ----- 269 A 55 5 90 ----- ----- 150 A3 3 4 12 ----- 19 A4 17 0 33 ----- 50 As ----- ----- ----- ----- ----- ----- Totals 183 17 288 488 Chi- Square df P 22.57 6 0.001 Cramer's V = 0.1521 Reset Calculate Note that 2 of your expected cell frequencies are smaller than 5. For a rows by columns chi-square test, at least 80% of the cells must have an expected frequency of 5 or greater, and no cell may have a an expected frequency smaller than 1.0. For a 2x2 table, the chi-square test is valid only if all expected cell frequencies are equal to or greater than 5. If this requirement is not met for a 2x2 table, use instead the Fisher Exact Probability Test. The Fisher Exact Test is also available for 2x3, 2x4, and 3x3. STOP: Enter your p-value and explain that means for your research question in #3 and interpret it in #4. 7. To determine how the observed values differ from expected, the calculator gives us a percentage deviations table, like the one shown below on left. This table tells us how different our observed values are from expected. We'll attempt to make some claims from this data next. Percentage Deviations B1 A1 +7.1% -14.6% B2 B3 -3.6% A -2.2% -4.3% +1.7% A3 -57.9% +504.3% +7% A4 -9.3% -100% +11.8% A5 Dry Dry-Wet Wet Wet-Dry Floodplain Savanna Limestone Gorge Mixed Savanna/ Woodland 7.1% -14.6% -3.6% -2.2% -4.3% 1.7% -57.9% 504.3% 7.0% -9.3% -100.0% 11.8% STOP: Type the numbers from the website's percent deviation table into your spreadsheet (example above) and then copy the percent deviation table from your spreadsheet into your lab report under #5. 8. Lastly, fill in what claims you can make from the percentage deviations table under #6 on the lab report. If your test rejected the null hypothesis (p <0.05), that means the data were statistically different, and animal species use of habitats change with the seasons. An example of this interpretation from my percent deviations table above would be: 3 Name: a. In the dry season, the floodplain was found to have 7.1% more individuals than expected. b. In the dry-wet season, the woodland was the only habitat with a positive deviation. c. In the wet season, the gorge had an overwhelmingly positive deviation from expectations. d. In the wet-dry season, the gorge has now flipped to a negative deviation. Synthesize these results in #7 to describe yearly animal behavior (if possible). Answer #8 to practice drawing claims from statistics. Finally, complete the reflection (#9). Congrats, you survived! 4 Name: Lab 5 Report 1. In your own words, what is a null hypothesis? What questions do you still have? [2 pts] 2. In your own words, what is a p-value? What questions do you still have? [2 pts] 3. What's your p-value for your species habitat-season relationship? [1 pt] 4. Does the p-value you provided support or reject the null hypothesis? Were the observed values significantly different than what would be expected due to chance? Can you make claims about the statistical significance of your data? [2 pts] Helpful reminders from the week 4 lab report: Null hypothesis: the distribution of observed numbers in your contingency table is NOT different from the expected numbers. In other words, you are testing the assumption that there is no difference in the groups tested. See the table below to learn more about what these p-values mean. P <0.05 REJECTS null hypothesis P>0.05 SUPPORTS the null hypothesis Observed values are significantly different Observed values are NOT significantly from what would be expected due to chance. different from what would be expected due to chance. Claims about the differences CAN be made Claims about the differences CANNOT be made 5. Percent Deviation Table [2 pts] 6. What claims can you make from the percentage deviations table? [4 pts, 1 pt for each season] Hint: You can only make claims about the significance of the data if you rejected your null hypothesis (p <0.05). If your null hypothesis was supported, briefly explain what that means regarding claims you make from your data. 7. Based on your graph from Lab 4 and your analyses above, how would you describe the movement of animals in the Gorongosa National Park over the course of the year? [2 pts] Note: your answer to this question depends on your p-value. 8. During Lab 4, you learned that graphs show patterns or trends in data, but claims should come from the statistical test. Analyze the following research question with their data visualization and statistical reporting. [4pts] Average Mass (grams) 900 800 700 600 500 400 300 200 100 Comparing American Marten mass between mainland and island populations. 0 Mainland Island Location Research Brief: J. Bristol Foster hypothesized in the 1960's that an animal species (martens, a small mammal) on an island would evolve over generations to be larger or smaller than individuals of the same species living on the mainland. To test this hypothesis, I examined natural history databases for American martens, a member of the weasel family, living in Alaska and the Alexander archipelago off the coast of Alaska. To measure size, I used the weight of martens on the mainland and martens on the islands. The average weight of these groups is graphed right. A t-test was used to determine whether there is a significant size difference between martens and the statistical test returned a p-value of 0.0322. a. What is the null hypothesis regarding martens' weights on the mainland and the island? b. Based on the graph, what trend do you see and is this trend significant according to the t-test? c. What claims can we make about Foster's hypothesis? d. What would you study next or what criticism of this study would you try to fix? 9. What was the main point of today's lab? Are there any concepts that you are still struggling with? If so, what are they? [1 pt]