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Lecture 2: Replication and pseudoreplication. This lecture will cover:. Experimental units (replicates) Pseudoreplication Degrees of freedom. Experimental unit. Scale at which independent applications of the same treatment occur Also called “replicate”, represented by “n” in statistics.

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## Lecture 2: Replication and pseudoreplication

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**This lecture will cover:**• Experimental units (replicates) • Pseudoreplication • Degrees of freedom**Experimental unit**Scale at which independent applications of the same treatment occur Also called “replicate”, represented by “n” in statistics**Experimental unit**Example: Effect of fertilization on caterpillar growth**Experimental unit ?**+ F + F - F - F n=2**Experimental unit ?**+ F - F n=1**Pseudoreplication**Misidentifying the scale of the experimental unit; Assuming there are more experimental units (replicates, “n”) than there actually are**Example 1.**Hypothesis: Insect abundance is higher in shallow lakes**Example 1.**Experiment: Sample insect abundance every 100 m along the shoreline of a shallow and a deep lake**Example 2.**What’s the problem ? Spatial autocorrelation**Example 2.**Hypothesis: Two species of plants have different growth rates**Example 2.**• Experiment: • Mark 10 individuals of sp. A and 10 of sp. B in a field. • Follow growth rate • over time If the researcher declares n=10, could this still be pseudoreplicated?**Example 2.**time**Temporal pseudoreplication:**Multiple measurements on SAME individual, treated as independent data points time time**Spotting pseudoreplication**• Inspect spatial (temporal) layout of the experiment • Examine degrees of freedom in analysis**Degrees of freedom (df)**Number of independent terms used to estimate the parameter = Total number of datapoints – number of parameters estimated from data**Example: Variance**If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Independent term method: Can the first data point be any number? Yes, say 8 Can the second data point be any number? Yes, say 12 Can the third data point be any number? No – as mean is fixed ! Variance is (y – mean)2 / (n-1)**Example: Variance**If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Independent term method: Therefore 2 independent terms (df = 2)**Example: Variance**If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Subtraction method Total number of data points? 3 Number of estimates from the data? 1 df= 3-1 = 2**Example: Linear regression**Y = mx + b Therefore 2 parameters estimated simultaneously (df = n-2)**Example: Analysis of variance (ANOVA)**A B C a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 What is n for each level?**Example: Analysis of variance (ANOVA)**A B C a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 df = 3 df = 3 df = 3 n = 4 How many df for each variance estimate?**Example: Analysis of variance (ANOVA)**A B C a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 df = 3 df = 3 df = 3 What’s the within-treatment df for an ANOVA? Within-treatment df = 3 + 3 + 3 = 9**Example: Analysis of variance (ANOVA)**A B C a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 If an ANOVA has k levels and n data points per level, what’s a simple formula for within-treatment df? df = k(n-1)**Spotting pseudoreplication**An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA (within-treatment MS). Is there pseudoreplication?**Spotting pseudoreplication**An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. Yes! As k=2, n=10, then df = 2(10-1) = 18**Spotting pseudoreplication**An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. What mistake did the researcher make?**Spotting pseudoreplication**An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. Assumed n=50: 2(50-1)=98**Why is pseudoreplicationa problem?**Hint: think about what we use df for!**How prevalent?**Hurlbert (1984): 48% of papers Heffner et al. (1996): 12 to 14% of papers

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