Preventing strokes Refer to Exercise 38. Do the data provide convincing evidence of a difference

Preventing strokes Refer to Exercise 38. Do the data provide...

Key Concepts

P-Value Interpretation

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed one assuming that the null hypothesis is correct. If the p-value is less than the significance level (? = 0.05), it indicates strong evidence against the null hypothesis, suggesting that there is a statistically significant difference among the treatments.

Significance Level

The significance level, often denoted by ? (with a common value of 0.05), is the threshold for determining whether a test statistic is extreme enough to reject the null hypothesis. It represents the probability of committing a Type I error, which is the error of rejecting the null hypothesis when it is actually true.

Degrees of Freedom

Degrees of freedom in the context of a chi-squared test refer to the number of independent values or quantities which can vary in an analysis. They are calculated based on the number of categories or groups minus certain constraints (such as the number of estimated parameters) and are used to determine the appropriate chi-squared distribution for assessing statistical significance.

Hypothesis Testing

Hypothesis testing is the process of formulating a null hypothesis, which typically states that there is no difference between groups or treatments, and an alternative hypothesis, which posits that a difference exists. In the context of comparing treatment effectiveness, it sets the stage for statistical decision-making about the impact of different treatments.

Chi-Squared Test for Homogeneity/Independence

The chi-squared test is used to compare categorical data across multiple groups to determine if observed differences in outcomes are statistically significant. It is particularly useful when assessing whether the distribution of responses (such as treatment outcomes) differs among several groups, operating under the assumption that any observed difference is due to random variation under the null hypothesis.

Transcript

00:03 All right, for this one, we're supposed to use all our data we've accumulated from prior questions about the effect of these medications on preventing strokes.

00:12 And we're supposed to form a kai score statistic at the point, at a 0 .05 level.

00:21 So there's some things we need to do first.

00:25 Oh.

00:26 So the null hypothesis, we need to state our hypotheses.

00:30 Null hypothesis is that, i think i already said it actually.

00:33 Didn't i say it before? yeah.

00:38 So the only positive is that all treatments are actually, i should try to be consistent with how we answer these.

00:52 So there is no difference in the proportion of strokes across treatments.

01:20 So there's no difference in the proportion of strokes across treatments.

01:28 So there's no difference in the proportion of strokes across treatments would be our null hypothesis, and then our alternate hypothesis would be that there is a difference in the proportion of strokes across treatments.

01:51 So our null hypothesis, essentially all these methods are equally successful in preventing strokes.

02:05 All right, so we did that.

02:09 And then next would be, so we did that.

02:13 So next, we would need to check the conditions.

02:29 So the conditions that we need to check, i don't know, i did colon sign there, we need to check randomness.

02:36 Well, the data came from a random experiment.

02:52 So we know that is random because they told us so.

02:57 Independent, well, due to the random assignment, we may treat each observation as independent.

03:28 So since people were randomly put into these treatment groups, whether or not someone has, does or not have a stroke, does not tell me anything about the next person because the next person is going to be another random person so random independent and then we have to do the large number check and indeed the expected outcomes are much much bigger than five so expected outcomes greater than are equal to five that definitely is true as well so all the conditions hold so we have our hypotheses we have our conditions so we can go ahead and calculate kai squared statistic so let's do that i like to kai squared in gold all right so kai squared um i'm going to just do some of these because there's a giant list and it's more like just plugging into a calculator so we take um the expected the difference between the expected and the observed and then we divide by the expected so i went ahead and did that so the first couple of terms would be 1399 minus 1443 .2 squared divided by 1443 .3 .2 plus the next one would be 250 minus 205 .8 squared divided by 205 .8.

05:12 So that's the first row.

05:14 I would just do this for the next.

05:18 Um you know all the next rows i would end my last two i'm going to leave it to you to do the middle ones because again it's just plug and chug the last two would be one four nine three minus one four four four point one divided by one four four point one plus one five seven minus two oh five point nine squared divided by two oh five point nine so you should have eight of these total and if you look here, notice my last one, 157 minus 205 .9.

05:55 So you can just check, what is the expected? 205 .9.

05:59 What did i subtract against it? 157.

06:03 And then i divided by the expected.

06:08 So i get 157 minus 205 .9 divided by 205 .1...

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