# ===================================================================== # 06 · T-TESTS: ONE, TWO, PAIRED — SCRIPT SKELETON # Ecological Statistics # # Boxed sections marked with rows of * * * * are written-answer questions # from the worksheet. Type your answer between the two closing rows of # stars, right in this script. # ===================================================================== # ── Part 1.1 · Setup ──────────────────────────────────────────────────── # FILLED IN — this is the same five lines at the top of every script. library(tidyverse) library(janitor) # round_half_up(), used inside summary_stats() library(car) # leveneTest() # Gives you theme_regular() and summary_stats(). # The file goes in a themes_functions/ folder next to your .Rproj. source("themes_functions/r_themes_and_functions.R") # Open the .Rproj first, then: pine_df <- read_csv("data/pine_data.csv") # ── Part 1.2 · Boxplot of all the raw needles ─────────────────────────── # FILLED IN — this is the plot pattern for the whole worksheet. # All 64 needles, before we fix anything. Jitter because 32 points # land on one x value and would otherwise hide each other. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: How many points are on this plot, and how many trees # did we actually climb? # points = ________ trees = ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 1.3 · Mean and SE of all the raw needles ─────────────────────── # FILLED IN — the mean ± SE pattern. Look hard at how SHORT these bars # are. n = 32 per side here, and that 32 is a lie. Part 1.7 fixes it. # stat_summary(fun = mean, geom = "point") + # stat_summary(fun.data = mean_se, geom = "errorbar", width = 0.2) + # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Do these two error bars overlap? Just from this # picture, would you expect a t-test to find a difference? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 1.4 · Average out pseudoreplication ──────────────────────────── # FILLED IN — the two-stage summary from Describing Your Data. # Everything below this line runs on p_df, not pine_df. 64 rows -> 8. # p_df <- # p_df # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: What is our true sample unit here -- needles? # branches? trees? tree sides? Why does that matter for the rest # of this worksheet? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 1.5 · Summary stats the long way ─────────────────────────────── # FILLED IN — the summarize() block you have written all semester. stats_df <- p_df %>% group_by(side) %>% summarize( n = sum(!is.na(needle_length_mm)), mean = mean(needle_length_mm, na.rm = TRUE), sd = sd(needle_length_mm, na.rm = TRUE), se = sd / sqrt(n), .groups = "drop" ) stats_df # ── Part 1.6 · Summary stats with summary_stats() ─────────────────────── # FILLED IN — same numbers, from the function you sourced in Part 1.1. summary_stats(p_df$needle_length_mm) # all 8 trees together p_df %>% group_by(side) %>% reframe(summary_stats(needle_length_mm)) # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write down n, mean, sd and se for each side, and # check them against stats_df. # shady: n = ______ mean = ______ sd = ______ se = ______ # sunny: n = ______ mean = ______ sd = ______ se = ______ # Do they match stats_df? ____________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: summary_stats() also gives you ci_lower and ci_upper -- # a 95% confidence interval. Look at the n column. How many degrees # of freedom did it use to build that interval, and why that number? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 1.7 · Boxplot and mean SE of the tree means ──────────────────── # YOUR TURN. This is Parts 1.2 and 1.3 again with ONE thing changed: # pine_df becomes p_df. Call them box_plot and mean_se_plot — Section 2 # reuses box_plot. No jitter needed now; there are only 4 points a side. # Then put raw_mean_se_plot and mean_se_plot side by side and compare. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Fill this in from your Part 1.3 and Part 1.6 output. # This is the most important table in the worksheet. # raw needles tree means # n per side __________ __________ # shady mean __________ __________ # sunny mean __________ __________ # shady se __________ __________ # sunny se __________ __________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: What happened to the MEANS when you averaged? What # happened to the STANDARD ERRORS? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Compare raw_mean_se_plot to mean_se_plot side by side # on your screen. Which one LOOKS like stronger evidence? Which one # is honest? Explain in one sentence why the raw version was not # actually more precise. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 1.8 · One line per tree ──────────────────────────────────────── # YOUR TURN. Same plot skeleton, but add group = team to the aes() and # use geom_line() + geom_point(). Call it slope_plot. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: The boxes in Part 1.7 overlapped a lot. Now look at the # lines. Do they all slope the same way? Write down what you see -- # this is the whole point of Section 4. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ===================================================================== # SECTION 2 · ONE-SAMPLE t-TEST # df = n - 1 = 3. # ===================================================================== # ── Part 2.1 · Just the shady side ────────────────────────────────────── # FILLED IN — plumbing, not a lesson. shady_df <- p_df %>% filter(side == "shady") shady_df # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write the question, prediction, and hypotheses. # Question (the biology, in plain English): # ____________________________________________________________ # Prediction (what you expect, and WHY -- the biology): # ____________________________________________________________ # ____________________________________________________________ # H0 (null hypothesis): # ____________________________________________________________ # Ha (alternative hypothesis): # ____________________________________________________________ # alpha = ____________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: How many trees are in this test, and how many means # did you estimate? So what is the df? # n = ________ means estimated = ________ df = ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 2.3 · Normality of the shady side ────────────────────────────── # YOUR TURN. Two things: a Q-Q plot (qqnorm() then qqline()) and # shapiro.test(). Both on shady_df$needle_length_mm. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Why do we check the shady side on its own rather than # p_df$needle_length_mm (both sides pooled)? What would pooling two # groups with different means do to the shape of the distribution? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 2.4 · One-sample t-test, two-tailed ──────────────────────────── # YOUR TURN. t.test() on one vector, with mu = the value you are # testing against. Save it as one_sample_model, then print it bare. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Copy the numbers out, and check the df against what # you predicted above. # t = ________ df = ________ p-value = ________ # sample mean = ________ mm 95% CI: ________ to ________ # Does the df match your answer above? ________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Does 15 mm fall inside or outside that confidence # interval? How does that relate to your p-value? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Your prediction above probably had a DIRECTION -- # shady needles are shorter than 15 mm. Write the one-tailed # hypotheses. # H0: mu ______ 15 # Ha: mu ______ 15 # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 2.5 · One-sample t-test, one-tailed ──────────────────────────── # YOUR TURN. The same call plus one argument: alternative = "less". # Save it as one_tailed_model. Compare the p-value to Part 2.4. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Fill in the comparison from your own output. # t df p-value # Two-tailed ________ ______ ________ # One-tailed ________ ______ ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: What happened to t? To df? To the p-value? (Look hard # at the relationship between the two p-values.) # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Look at the confidence interval in the one-tailed # output. What is the lower bound, and why does that make sense? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 2.6 · Critical values, one vs two tailed ─────────────────────── # YOUR TURN. Two calls to qt(), both at df = 3. One for the two-tailed # cutoff, one for the one-tailed. Which is closer to zero? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Which cutoff is closer to zero? So which test is easier # to get a significant result from, and why? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 2.7 · The wrong tail ─────────────────────────────────────────── # YOUR TURN. Part 2.5 again with alternative = "greater". Same data, # same t. Watch what the p-value does. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Same data, same t. What is the p-value now? What # does that tell you about WHEN you are allowed to choose the tail? # p-value = ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # BONUS (finish early): A one-tailed test is defensible when a # result in the opposite direction leads to the SAME action as no # result at all. For our pine needles, is 'sunny is longer' AND # 'shady is longer' both interesting biology? So which should we use? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN (2.8 interpretation): Do you reject or fail to reject # H0 at alpha = 0.05? In biological terms, what does this say about # shady-side needles? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write the result the way it would appear in a paper. # "A two-tailed, one-sample t-test at alpha = 0.05 showed _______ # ________________________________________________________ # t(______) = ________, p = ________." # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ===================================================================== # SECTION 3 · TWO-SAMPLE t-TEST # df = n1 + n2 - 2 = 6. Welch's df is fractional. # ===================================================================== # ── Part 3.1 · Question, prediction, hypotheses ───────────────────────── # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write all five down BEFORE you run anything. # Question (the biology, in plain English): # ____________________________________________________________ # Prediction (what you expect to see, and WHY -- the biology): # ____________________________________________________________ # ____________________________________________________________ # H0 (null hypothesis): # ____________________________________________________________ # Ha (alternative hypothesis): # ____________________________________________________________ # alpha = ____________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Your prediction probably had a direction ('sunny # needles are longer'). Is your Ha directional too, or is it just # 'they differ'? Which one does t.test() run by default? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 3.2 · Normality by group ─────────────────────────────────────── # YOUR TURN. group_by(side), then summarize() with # shapiro.test(needle_length_mm)$p.value. Test each side separately. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: What is the null hypothesis of the Shapiro-Wilk test? # Do you WANT a significant or a non-significant result here? # H0 of Shapiro-Wilk: __________________________________________ # I want p to be: __________________________________________ # My result: shady p = ____________ sunny p = ____________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 3.3 · Equal variances ────────────────────────────────────────── # YOUR TURN. leveneTest(), formula interface: y ~ factor(group). # It needs the grouping variable as a factor. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: What is the null hypothesis of Levene's test? Based on # your p-value, should you use the standard (Student's) t-test or # Welch's? # H0 of Levene's test: _________________________________________ # My p-value: ____________ # Test I will use, and why: ____________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 3.4 · Two-sample t-test ──────────────────────────────────────── # YOUR TURN. t.test() with the formula interface this time: # t.test(y ~ group, data = ..., var.equal = ...) # Run it twice — var.equal = TRUE, then FALSE. # Call them two_sample_model and welch_model. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Copy the numbers out of the output by hand. # t df p-value # Standard (Student's) ________ ________ ________ # Welch's ________ ________ ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Welch's df is not a whole number, and it is smaller # than the standard df. Why? What did Welch's buy you in exchange? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN (3.5 interpretation): Do you reject or fail to reject # H0 at alpha = 0.05? What does the p-value actually mean here? # (Careful -- it is a statement about the data GIVEN H0, not about # whether H0 is true.) In biological terms, what does this result # say about pine needles? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write the result the way it would appear in a paper. # The number in parentheses after t is the degrees of freedom -- # always report it. # "A two-tailed, two-sample t-test at alpha = 0.05 showed ______ # ________________________________________________________ # t(______) = ________, p = ________." # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 3.6 · Two-sample t by hand (bonus) ──────────────────────────── # YOUR TURN. sunny_df is below; shady_df is from Part 2.1. sunny_df <- p_df %>% filter(side == "sunny") # shady_df is from Part 2.1 # Build the pooled SD, then the t-statistic. Check it against Part 3.4. # ===================================================================== # SECTION 4 · PAIRED t-TEST # df = n_pairs - 1 = 3. The 8 numbers became 4 differences. # ===================================================================== # ── Part 4.1 · Question, prediction, hypotheses ───────────────────────── # Notice H0 and Ha are now about ONE number: the mean difference. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: # Question (the biology, in plain English): # ____________________________________________________________ # Prediction (what you expect, and WHY): # ____________________________________________________________ # ____________________________________________________________ # H0 (in terms of the mean difference between the two sides # of a tree): # ____________________________________________________________ # Ha: # ____________________________________________________________ # alpha = ____________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 4.2 · Paired t-test ──────────────────────────────────────────── # FILLED IN — the reshape is plumbing. One row per tree, two columns. p_wide_df <- p_df %>% pivot_wider(names_from = side, values_from = needle_length_mm) p_wide_df # YOUR TURN. t.test() on the two columns with paired = TRUE. # Shady first, then sunny — the same order the two-sample test used. # Call it paired_model. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Copy the numbers out, and note the df. # t = ________ df = ________ p-value = ________ # mean difference = ________ mm (this is ________ minus ________) # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 4.3 · Compare the two tests ──────────────────────────────────── # YOUR TURN. Pull the p-value out of each model with $p.value. # Same 8 numbers. Why are they so different? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Fill in the table from your own output. # Test t df p-value # Two-sample ________ ______ ________ # Paired ________ ______ ________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Both tests used the SAME 8 numbers and both estimated # the SAME difference in means. So why are the p-values so # different? (Think about where tree-to-tree variation ends up # in each test.) # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: The paired test has HALF the degrees of freedom -- 3 # instead of 6. Fewer df makes significance HARDER to reach, not # easier. So how did it still do better? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 4.4 · Paired = one-sample on the differences ─────────────────── # YOUR TURN. mutate() a difference column on p_wide_df (sunny - shady), # then run a plain one-sample t.test() on it against mu = 0. # Call it diff_model. Compare t, df and p to paired_model. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Compare diff_model to paired_model. Write down the t, # the df and the p-value for each. What do you notice? # paired_model: t = ________ df = ______ p = ________ # diff_model: t = ________ df = ______ p = ________ # What I notice: _________________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: So what is paired = TRUE actually doing for you? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: This is the answer to the degrees-of-freedom question. # The two-sample test on these same measurements had df = 6. This # one has df = 3. Explain why, in terms of how many numbers # diff_model is actually working with. # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN (4.5 interpretation): Do you reject or fail to reject # H0 at alpha = 0.05? Which test -- two-sample or paired -- is the # CORRECT one for how we collected this data, and why? In # biological terms, what does the paired result say about pine # needles? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Write the result the way it would appear in a paper. # "A two-tailed, paired t-test at alpha = 0.05 showed __________ # ________________________________________________________ # t(______) = ________, p = ________." # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # BONUS (finish early): The shady side 'controls for' everything # about that particular tree. Which assumption from Section 3 does # this design let you stop worrying about? And which variable's # normality should you now be checking? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ===================================================================== # SECTION 5 · TAKE-HOME EXTENSION — due Sept 28 # You choose the tests and justify them. No hints past this point. # ===================================================================== # ── Part 5.1 · Take-home extension ────────────────────────────────────── mice_df <- read_csv("data/mice_weights.csv") mice_df %>% group_by(location) %>% reframe(summary_stats(mass_g)) # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN (Task A -- one-sample question): Do island mice (Sidney # Island) differ from a 'typical' continental deer mouse body mass # of 19 g? This is Section 2 again, with a different mu and dataset. # Question: ______________________________________________ # Prediction: ______________________________________________ # ______________________________________________ # H0 = ______________________________________________ # Ha = ______________________________________________ # Test I will use, and why: ____________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 5.2 · Task A — one-sample question ───────────────────────────── # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Result and interpretation for Task A. # Result: t(______) = ________, p = ________ # Interpretation: # ____________________________________________________________ # ____________________________________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN (Task B -- two-sample question): Does mouse body mass # differ between the island (Sidney Island) and mainland # (Vancouver) populations? Run Levene's test again yourself before # you pick standard vs. Welch's. # Question: ______________________________________________ # Prediction: ______________________________________________ # ______________________________________________ # H0 = ______________________________________________ # Ha = ______________________________________________ # Levene's p = ____________ # Test I will use, and why: ____________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 5.3 · Task B — two-sample question ───────────────────────────── # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Result and interpretation for Task B. # Result: t(______) = ________, p = ________ # Interpretation: # ____________________________________________________________ # ____________________________________________________________ # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # YOUR TURN: Could you have run a PAIRED t-test on these mice? # Why or why not? # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ── Part 5.4 · Final figure ───────────────────────────────────────────── # ===================================================================== # END # =====================================================================