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## Express your answer to two significant figures and include the appropriate units.?

l = length of the metal block = 17.1 cm

w = width of the metal block = 3.2 cm

h = height of the metal block = 4.8 cm

V = volume of the metal block = to be determined

ρ = density of the metal block = to be determined

m = mass of the metal block = 1.5 kg

V = lwh

V = (17.1 cm)(3.2 cm)(4.8 cm)

V = 262.656 cm³

ρ = m/V

ρ = (1.5 kg)/(262.656 cm³)

ρ = 0.0057 kg/cm³ = 5.7 g/cm³ ANSWER

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Express your answer to two significant figures and include the… – SubmitPrev Previous Answers Request Answer X Incorrect; Try Again; 5 Attempts Remaining Part B 77 Ft/s Express Your Answer To Two Significant Figures And Include The Appropriate UnitsDo not include units in your numerical answers. The appropriate units should be shown just after the entry box. Eduspace uses the "E" notation for "times 10 to the" in scientific notation. For example, 56000, with 3 significant figures would be expressed 5.60 × 104, and should be entered in…In this case, all numbers including 2450 and between 2450 and 2550 (but not including 2550) are rounded to 2500 when rounding to the nearest 100. If you want to round off 2456 to 2 significant digits, you will need to write your answer with an accuracy of 100 units, but using only two figures.

**Physics CH 0: General Introduction (9 of 20) Multiplying with Uncertainties in Measurements** – .

**Chemistry Lesson: Significant Digits & Measurements** – .

**Unit 2 Module 4 part 1** – In this next module, we're going to look

at another measure of association called the relative risks.

Instead of subtracting the risks in the different groups, what if we instead

divided them to make a ratio, that's called a relative risk.

There are several measures of relative risk and relative risks appear very

frequently in the medical literature. If you take two incidents rates and

divide them, that's called a rate ratio. You'll also see something in the

literature called a hazard ratio. A hazard ratio is also a ratio of

incidence rates. It's just a ratio of what we call

instantaneous incidence rates or hazard rates.

Hazard ratios actually come from a regression technique that we're going to

talk about much later in the course. But for now, you can just interpret them

exactly the same way you would interpret a rate ratio.

If you divide two cumulative risks, or two prevalences, we call that a risk

ratio. Now some people might call the ratio of

two prevalences a prevalence ratio, to distinguish it, but I tend to lump them

together and call them all a risk ratio because cumulative risks and prevalences

are both just proportions. Another measure of relative risk I'm

going to talk about a great deal this week is something called an odds ratio.

An odds ratio is a ratio of two odds and I'm going to have to tell you what an

odds is in a minute because you may not really understand what an odds is

already. But the odds ratio comes up a lot in the

medical literature for two reasons. The odds ratio is the only value measure

of relative risk for case control studies.

That's one reason. The other reason that odds ratios come up

a lot, is that a lot of researchers want to do a regression technique called

logistic regression. It's a multivariate regression technique

that's used when you have a binary outcome, and you want to adjust for

confounders. It turns out that logistic regression

gives you out odds ratios. So even if you're doing a covert study,

or a cross sectional study, if you use logistic regression, you're going to get

out odds ratios. So, odds ratios are all over the

literature. But as we're going to see in this module

and the next module, they're a little bit of a funny measure and sometimes they're

tricky to interpret. [BLANK<u>AUDIO] So how do we interpret

relative risk?</u> So the rate ratio and the hazard ratio,

you can interpret as the percent increase or decrease in the rate of the outcome.

The risk ratio is the percent increase or decrease in the risk or prevalence of the

outcome. The odds ratio is the percent increase or

decrease in the odds of the outcome, and again I'll fill you in on what an odds is

in a minute. If you have a relative risk that's equal

to 1.0, that means that the risks in the two groups are equal, because, say if the

risk is 20% in one group and 20% in the other group, if you divide 20% by 20%

you're going to get 1.0. So that represents where there's no

difference between the groups, we call that the null value.

If you have a relative risk that's less than one, that indicates a protective

effect. Whatever was in the numerator is

protecting you from the outcome. If you have a relative risk that's

greater than one, that indicates a harmful effect, an increased risk.

Let's calculate the rate ratio first starting with the GI event data from the

Vioxx study we've been talking about. Incident rates were 2.1 and 4.5 again.

All we have to do here to make a rate ratio is to divide the two incidents

rates. And notice because we're dividing two

rates, the units here, the 100 person-years actually cancels out.

And the rate ratio doesn't have any units, it's unitless.

We come out with a rate ratio of 0.46. The interpretation is that Vioxx reduces

the rate of GI events by 54%. I got that 54% simply by subtracting 1

minus 0.46. So that's a 40, 54% reduction in the rate

of GI events. If I want to calculate the risk ratio for

this same data, I'm just going to divide the risk in the Vioxx group by the risk

in the Naproxen group. Interestingly I get exactly the same a

number here, 0.46. So in this case the rate ratio and the

risk ratio are identical because, remember, the person-years of follow up

was exactly the same in both groups. So, if you're following people for the

same amount of time and the risk ratio and the weight ration would come out to

be the same. However, if there was a differential

follow up between the groups then this wouldn't be the case.

Because if one group was followed for longer than the other group, that would

increase the risk in the group that was followed longer, and it would change the

risk ratio compared with the, the rate ratio.

Now I want to read to you what they put in the pay, in the paper's abstract.

During a median follow-up of 9 months, 2.1 confirmed GI events per 100

patient-years occurred with the rofecoxib, that is the Vioxx group, as

compared with 4.5 per 100 patient years with naproxen.

Now they're going to give the relative risk.

The relative risk is 0.5 so they rounded up from 0.46 to, to 0.5.

95% confidence interval, 0.3 to 0.6 and it's highly statistically significant.

I want to spend just a minute here to walk you through what that confidence

interval means. We're going to spend a lot of time on

confidence intervals in an upcoming week of the course.

But I want to just give you sort of a conceptual understanding of what that

confidence interval means here. So the relative risk that we calculated

in the study, what we call the point estimate.

What we calculated in the study was exactly 0.46 rounded up to .5.

So that indicates a halving of risk. But that's just what we saw in the study.

And of course, whenever we do a study, we know that there's going to be some

uncertainty. In other words the true effect of Vioxx

on GI events might not be exactly 0.5. We understand that when we do a study,

there's uncertainty. We have to put a margin of error around

our estimate. That margin of error is what we call the

95% confidence interval. So the 95% confidence interval here is

0.3 to 0.6. That gives us a plausible range of values

for the true effect. The real effect of Vioxx on on GI events

we believe is somewhere between 0.3 and 0.6.

Another way to interpret it is we say we can be 95% confident that the true effect

of Vioxx verus naproxen is between a 40% and 70% reduction in the rate of GI

events. Where did that 40% and 70% come from?

The upper limit of the confidence interval is a relative risk of 0.6.

That's a 40% reduction in risk. The lower limit in the confidence

interval is a relative risk of 0.3. That's a 70% reduction in risk.

Now let's move to the heart attack data from the study.

What's the rate ratio going to be here. The rate ratio will be the rate in the

Vioxx group divided by the rate in the Naproxen group.

When you do that division it comes out to be 4.2, if the risk ratio is actually

identical, as I said before, because the groups should follow for equal amounts of

time. The risk ration and the rate ratio are

the same. How do we interpret these results?

We would interpret this that Vioxx increases the rate or risk of heart

attacks by fourfold, or you would say by 320%.

But let's see what the authors actually put in the abstract here.

So, here's the abstract from the paper. The incidence of myocardial infarction,

heart attack, was lower among patients in the naproxen group than among those in

the Vioxx group. 0.1% versus 0.4%.

We already saw that number in the last module.

Relative risk 0.2, 95% confidence interval, 0.1 to 0.7.

So notice that when the authors reported the data on GI events, they put the

naproxen in the denominator. They put the naproxen as the reference

group, but when they report the data on heart attacks, they're flipping things

around. They're making Vioxx the reference group.

They're putting Vioxx in the denominator. And you can probably think of the reason

that they might have wanted to do that. So there are actually two funny things

going on here, as I alluded to in the last module.

So firstly, they reported risks rather than rates.

For the primary outcome GI events, remember they actually reported the

incidence rates, they reported the risks here, likely because the risk difference

here looks smaller than the rate difference.

Secondly, they flipped the relative risk. They divided the risk in the Naproxen

group by the risk in the Vioxx group rather than the other way around.

So remember the risk in the Vioxx group was 0.42.

The risk in the Naproxen group is 0.1. If you divide 0.1 by 0.42 you get a

relative risk of 0.24 and they rounded down in the abstract to 0.2.

They rounded to one decimal place. So they twisted the message here, right.

They twisted the message to make it look like naproxen is protective rather than

that Vioxx is harmful. And technically, they haven't done

anything wrong mathematically. The numbers are correct.

But flipping the relative risk of course completely shifts the take home message,

and people missed it. Not one person wrote a letter to the

editor of the New England Journal, after this article was published, questioning

the author's logic here. People bought into the story that the

authors were trying to sell, that Naproxen is this miracle heart protective

drug. instead of that Vioxx is a dangerous

drug. Here's the, what they wrote in the

paper's conclusion. I showed you this in the teaser earlier.

Thus, our results are consistent with the theory that Naproxen has a coronary

protective effect. And highlight the fact that rofecoxib

Vioxx does not provide this type of protection.

The finding that naproxen therapy was associated with a lower rate of

myocardial infarction needs further confirmation in larger studies.

Of course, naproxen is the reference group, the control group in this study.

So every comparison really outta be Vioxx versus naproxen, not naproxen versus

Vioxx. But they totally put the focus on

naproxen to take the focus away from Vioxx and it worked.

in fact it turns out that naproxen has a very small, if any cardio protective

affect. So here was clear evidence sitting in the

literature in 2000 that Vioxx raises the risk of cardiovascular events.

Some people think that if if people had actually looked at this study more

carefully and kind of questioned the reasoning of the authors here, that Vioxx

might have actually been pulled off the shelves sooner than 2004.

The data were sitting here showing us that there's this cardio harmful effect

of nap, of Vioxx. And there's actually a lot of backstory

here that I'm not telling you. There was actually some data omitted from

the New England Journal of Med, of Medicine paper.

if you're interested in all of this, you can Google it, you can find out more

about this story. people made a big deal about the fact

that some of the heart, actually three heart attacks that probably should have

been included in that New England Journal paper that weren't.

But in fact, that data that was already available in the paper really told the

whole story. It's just that the authors had managed to

spin that story and people bought it. So the bigger lesson here, is that it's

very important when you're reading a medical paper to always look at the data

yourself. And then interpret the data yourself.

Because you allow the authors to lead you through their interpretation of the data,

you can easily get sucked into the story they want to tell you.

And that's what happened here. .