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ARE OBSERVATIONS OBTAINED DIFFERENT?

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ARE OBSERVATIONS OBTAINED DIFFERENT? You use different statistical tests for different problems. We will examine some basic tests ( χ 2, t-test, Regression, ANOVA, ANCOVA, χ 2 ) We expect you to use these basic tests in your research. Your research project should not be so complicated that more advanced tests are required. Always state your hypothesis – what you are testing.

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BASIC PREMISE OF STATISTICAL TESTING: You observe 60 heads. Is the coin fair? sd away from mean = (60 – 50)/5 = 2 sd 2 sd is 5% chance, but in one direction so 2.5% chance (5%/2) Proportion of heads Frequency Toss a coin 100 times A fair coin: x = 50 heads sd = 5 heads (√(½ x ½ x 100)) What if you set the probability to claim it to be unfair to be 5%? What if you set the probability to claim it to be unfair to be 25%? NULL HYPOTHESISACCEPTEDREJECTED TRUECORRECTTYPE I ERROR FALSETYPE II ERRORCORRECT NULL HYPOTHESIS Null Hypothesis: The coin is fair

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NONPARAMETRIC TESTS: (data does not have to be normally distributed) strata RED 8/10 RED 1/11 RED 1/13 #1#2 #3 2 CONTINGENCY TABLE: STRATA #1 #2#3 SPECIES RED NOT RED 8 1 1 10 2 10 12 24 10 11 13 34 Expected for each cell = (R x C)/TOTAL (2.94 ) (3.24) ( 3.82) (7.06) (7.76) ( 9.18) (O – E) 2 2 = E 8.71 + 3.63 +1.55 +.65 + 2.08 +.87 = 17.49 = P < 0.001; df = (r-1)(c-1) = 2 Data must be counts and you test proportional distribution of counts. Null hypothesis: no difference in proportion of red among strata

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2 CONTINGENCY TABLE: Make a spreadsheet with table categories and counts in each, and then have MYSTAT use as frequencies (Data … Case weighting … By frequencies) Depending on table, use One-way frequency tables (one category – e.g., tree type) or Tables (more than one category – e.g., tree type and strata) in Analyze in MYSTAT

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PARAMETRIC TESTS (data is normally distributed) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 t = [(0.71)(1.41)]/.214 = 4.68 P < 0.005, degrees of freedom = 6 T-TEST: (x 1 - x 2 )√n 1 n 2 /(n 1 + n 2 ) √[(n 1 – 1)s 1 2 + (n 2 – 1)s 2 2 ]/(n 1 + n 2 – 2) t = Data do not have to be counts. Easier to see differences (more powerful) than nonparametric statistics. Null hypothesis: no difference in proportion of red between strata #1 and #2. Proportion red Frequency

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T-TEST: Use Hypothesis testing in Analyze in MYSTAT for means

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PARAMETRIC TESTS (data is normally distributed) EVEN MORE POWERFUL IF A PRIORI BASIS TO PAIR OBSERVATIONS. strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 P < 0.001, degrees of freedom = n-1 = 3 PAIRED T-TEST: Pairs: 0.5 – 0 = 0.5; 1.0 – 0 = 1.0; 1.0 - 0.33 = 0.67; 0.67 – 0 = 0.67 mean = 0.71, sd = 0.21 t = 0.71/(0.21/√4) = 6.76 Data do not have to be counts. Easier to see differences (more powerful) than nonparametric statistics. Null hypothesis: no difference in relative abundance of red between strata #1 and #2 for matched plots based on similarity.

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PARAMETRIC TESTS (data is normally distributed) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 Strata #1: mean = 2.0, sd = 0.82, n = 4 Strata #2: mean = 0.25, sd = 0.5, n =4 t = [(2 – 0.25)(1.41)]/ 0.68 = 3.63 P < 0.01, degrees of freedom = 6 T-TEST: Data do not have to be counts. Easier to see differences (more powerful) than nonparametric statistics. Null hypothesis: no difference in absolute abundance of red between strata #1 and #2. Now use numbers not proportions.

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strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 STATISTICAL TESTS Null hypothesis: there is no relationship between red vs. blue + green in plots. REGRESSION ANALYSIS: 5 RED = 2.33 – 0.75(BLUE or GREEN) r 2 = 0.75, r = -0.88 Degrees of freedom = 12 – 2 = 10 P < 0.001

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REGRESSION ANALYSIS: Use Regression … Linear … Least squares in Analyze in MYSTAT Select dependent (y) and independent (x) variables

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WHAT IF MULTIPLE COMPARISONS OF A CATEGORY (ANOVA) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 Three possible t-test comparisons: #1 vs. #2 #1 vs. #3 #2 vs. #3 PROBLEM: As number of comparisons increases, the likelihood of finding at least one significant difference by chance increases. ANOVA takes this into account to compare differences in mean values. F = 19.75 df = 2, 9 (strata -1, samples – strata) p < 0.001 PARAMETRIC TESTS (data is normally distributed) Null hypothesis: no difference in relative abundance of red among all strata. 1-WAY ANOVA:

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ANOVA: Use Analysis of variance … Estimate model in Analyze in MYSTAT Select continuous dependent (y) variable and categorical independent (x) variables

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MULTIPLE COMPARISONS (ANOVA): (Which specific differences are significant?) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 Post –hoc analysis: Must compensate for number of comparisons and the fact that a difference is already known to be significant. Bonferroni test : (t-test adjusted for # of comparisons) #1 vs. #2 – p < 0.001 #1 vs. #3 – p < 0.001 #2 vs. #3 – p < 1.0

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ANOVA – POST HOC: (cannot do with MYSTAT, but will with SYSTAT) Use Analysis of variance … Estimate model … Hypothesis test in Analyze in SYSTAT

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MULTIPLE COMPARISONS (ANOVA): (several independent categorical variables) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 DISTANCE FROM EDGE near far Strata: F = 15.65; df = 2,6; p < 0.001 Distance: F = 0.12; df = 1,6; p < 0.74 Strata X Distance Interaction: F = 0.51; df = 2,6; p < 0.63 Null hypothesis: no difference in relative abundance of red between strata and with distance into the woods. TWO-WAY ANOVA: COULD HAVE N-WAY ANOVA, YOUR PROJECT SHOULD NOT EXCEED A 2-WAY.

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SEASONABCmean I 1232 II 2222 III 3212 mean222 2 LOCATION NO MAIN EFFECTS (SEASON or LOCATION – no differences) INTERACTION IS SIGNIFICANT (greatest at A:III and C:I) THE INTERACTION TERM’S MEANING (no variety)

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SEASONABCmean I 1232 II 4565 III 7898 mean456 5 LOCATION MAIN EFFECTS (SEASON or LOCATION -- differences) NO INTERACTION (highest always in C and III) (wider variety) THE INTERACTION TERM’S MEANING

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MULTIPLE COMPARISONS (ANCOVA): (several independent variables: one categorical and one continuous ) strata RED 0.79 + 0.25 RED 0.08 + 0.17 RED 0.08 + 0.17 #1#2 #3 DISTANCE FROM EDGE near far ANCOVA: Blue + Green: F = 36.10; df = 1,9; p < 0.0002 Distance: F = 0.78; df = 1,9; p < 0.40 Interaction (slope): F = 0.08; df = 1,8; p < 0.08 Null hypothesis: no difference in relative abundance of red with blue + green and distance into the woods (assume equal slopes). COULD HAVE N-WAY ANCOVA,

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ANCOVA: Use Analysis of variance … Estimate model in Analyze in MYSTAT. In SYSTAT use General linear model … Estimate model in Analyze Select continuous dependent (y) variable and categorical independent (x 1 ) variable and covariate (x 2 ). In SYSTAT, create interaction term to test slope.

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DATA TRANSFORMATIONS (can normalize data or make it continuous so parametric statistics can be used, or make data linear for regression) Data are not always normally distributed, but a transformation may make it normal (e.g., log). If it cannot be normalized then must use non-parametric statistics (less powerful). Data are not always continuous, percentages or proportions are not continuous because they cannot be less than 0 or greater than 100 or 1. To make them continuous from 0 to infinity or –infinity to +infinity, you can use transforms: arcsine transform = arcsin proportion; logarithmic transform = log(proportion)* logit transform = log (proportion/1-proportion)*. This stretches both tails and compresses the peak to approximate a continuous normal distribution. * If some proportions = 0 or 1, then add a small constant to all values (e.g, 0.001) Data for regression are not always linear, various transformations, especially log x, log y or both, can transform a curve into a straight line. What do logarithmic transforms imply about the linear function?

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DATA TRANSFORMATIONS Use Data … Transform … Let in MYSTAT.

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ARE OBSERVATIONS OBTAINED DIFFERENT? Different statistical tests for different problems. You will use these basic tests in your research ( χ 2, t-test, Regression, ANOVA, ANCOVA ) Your research project should not be so complicated that more advanced tests are required. Always graph your data and state your hypothesis.

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USE MYSTAT WITH DATA FILES CREATED LAST WEEK (be sure to set 6 decimal places -- Edit … Options … Output in MYSTAT so p values are exact) Meadow vole (Microtus pennsylvanicus) Yellowbellied marmot (Marmota flaviventris) UNDERC-WEST (National Bison Range)

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Does snap-trapping lead to a sex bias in Microtus? What is the relationship between length and mass for Microtus? (hint: need to useData … Transform … Let) Do Microtus and Marmota exhibit similar length and mass growth relationships? (hint: think about question above) Does Marmota mass vary with month? Explain ecologically what you see. Does reproductive status of female Microtus differ with mass? Why do you observe this? (hint: need to use Data … Select cases) Does the reproductive status of male and female Microtus with mass differ? Due in two weeks! WITH MYSTAT ANSWER THESE QUESTIONS: (you will use χ 2, regression, t-test, 2-way ANOVA, ANCOVA)

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