There is one chemical for plants; in absence of it (control) all 3 of them live. In presence 5 of 6 die, only 1 lives. So how can we show if it is significant or not? It may be basic but I appreciate your help.
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$\begingroup$ Please don't just reject proposed solutions without stating what bothers you. If anything it will prevent more people from contributing. As things stand, I think the exact test is your best option. $\endgroup$– servais daligouMar 31, 2015 at 19:22
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2 Answers
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I suggest to use the Fisher's Exact test in order to test if there is a statistically significant difference between the proportions of survivors in the two samples (treatment and control).
Check http://en.wikipedia.org/wiki/Fisher%27s_exact_test
The contingency table will look like:
................. Alive | Dead | Row Total
Treatment 1 | 5 |6
Control..... 3 | 0 | 3
Col Total.. 4 | 5 | 9
answered Mar 31, 2015 at 18:58
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$\begingroup$ Thnak you, i looked at wikipedia and it says, it calculates exact p value and it is good for small data. I can't accept as answer but i appreciate your help. $\endgroup$– xcvbnmMar 31, 2015 at 19:09
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$\begingroup$ That's OK, but I don't understand why Fisher's test doesn't suit your needs... $\endgroup$ Mar 31, 2015 at 19:12
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$\begingroup$ It suits and i just said that. I dont know the details of that test, I will definitely look more in detail, but according to introduction in wikipedia, it suits just as you suggested. Edit: I said I cant accept your answer because this site didnt let me because i needed some reputation. but now it let me, i dont know why $\endgroup$– xcvbnmMar 31, 2015 at 19:18
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Five of all the plants in the experiment died. If the experiment was well-run, the only difference between the two groups under the null hypothesis was the randomized choice of $6$ out of $9=3+6$ plants that were placed into the treatment group.
An accurate model puts nine balls into an urn, one for each plant: five of them black (to represent plants that eventually died) and four of them white (the plants that lived). The randomization selected six of those balls from the urn, leaving just three to be in the control group.
The probabilities of all the possible outcomes are elementary to compute: the chances of drawing $5$, $4$, $3$, and $2$ black balls in the treatment group are $1/21$, $5/14$, $10/21$, and $5/42$, respectively. (This is a hypergeometric distribution.)
An "extreme" result contrary to the null hypothesis--assuming the only plausible effect of the treatment would be to make the plants more likely to die--would be the appearance of a large proportion of black balls in the treatment group. There are three critical regions for such a "one-sided" test:
$C_5$ is the event that all five black balls were randomized into the treatment group. Its probability is $1/21 \approx 4.8\%$.
$C_4$ is the event that four or more black balls were in the treatment group. Its probability is $1/21 + 5/14 \approx 40.5\%$.
$C_3$ is the event that three or more black balls were in the treatment group. Its probability is $1/21 + 5/14 + 10/21 \approx 88.1\%$.
(There are no more critical regions because at least two of the black balls had to be in the treatment group.)
The smallest of these critical regions (in the sense of its probability) into which the experimental outcomes belong is $C_5$. Its chance, $4.8\%$, is the p-value. If your threshold for significance is larger than this, you may conclude the results are significant. That is, you observed an outcome that (a) was relatively rare if only chance determined the differences between control and treatment groups yet (b) would be relatively likely if the plants in the treatment group were adversely affected by the chemical.
This analysis is called Fisher's Exact Test. In R it could be conducted exactly with a command like
fisher.test(cbind(c(3,1), c(0,5)))$p.value
or approximately with the command
chisq.test(cbind(c(3,1), c(0,5)), simulate.p.value=TRUE)
The concepts underlying the probability model and hypothesis testing are discussed--from several points of view and at greater length--at What is the meaning of p values and t values in statistical tests?, inter alia.
