Significance levels in statistics are a crucial component of hypothesis testing. However, unlike other values in your statistical output, the significance level is not something that statistical software calculates. Instead, you choose the significance level. Have you ever wondered why? Show - Evidentiary Standards in the Courtroom
- Significance Levels as an Evidentiary Standard
- Changing Significance Levels
- Increasing the Significance Level
- Decreasing the Significance Level
- What Is Statistical Significance?
- How is it calculated?
- How Do Companies Use It?
- What Mistakes Do People Make When Working with Statistical Significance?
In this post, I’ll explain the significance level conceptually, why you choose its value, and how to choose a good value. Statisticians also refer to the significance level as alpha (α). First, it’s crucial to remember that hypothesis tests are inferential procedures. These tests determine whether your sample evidence is strong enough to suggest that an effect exists in an entire population. Suppose you’re comparing the means of two groups. Your sample data show that there is a difference between those means. Does the sample difference represent a difference between the two populations? Or, is that difference likely due to random sampling error? That’s where hypothesis tests come in!Your sample data provide evidence for an effect. The significance level is a measure of how strong the sample evidence must be before determining the results are statistically significant. Because we’re talking about evidence, let’s look at a courtroom analogy.
## Evidentiary Standards in the CourtroomCriminal cases and civil cases vary greatly, but they both require a minimum amount of evidence to convince a judge or jury to prove a claim against the defendant. Prosecutors in criminal cases must prove the defendant is guilty “beyond a reasonable doubt,” whereas plaintiffs in a civil case must present a “preponderance of the evidence.” These terms are evidentiary standards that reflect the amount of evidence that civil and criminal cases require. For civil cases, most scholars define a preponderance of evidence as meaning that at least 51% of the evidence shown supports the plaintiff’s claim. However, criminal cases are more severe and require stronger evidence, which must go beyond a reasonable doubt. Most scholars define that evidentiary standard as being 90%, 95%, or even 99% sure that the defendant is guilty. In statistics, the significance level is the evidentiary standard. For researchers to successfully make the case that the effect exists in the population, the sample must contain a sufficient amount of evidence. In court cases, you have evidentiary standards because you don’t want to convict innocent people. In hypothesis tests, we have the significance level because we don’t want to claim that an effect or relationship exists when it does not exist. ## Significance Levels as an Evidentiary StandardIn statistics, the significance level defines the strength of evidence in probabilistic terms. Specifically, alpha represents the probability that tests will produce statistically significant results when the null hypothesis is correct. Rejecting a true null hypothesis is a type I error. And, the significance level equals the type I error rate. You can think of this error rate as the probability of a false positive. The test results lead you to believe that an effect exists when it actually does not exist. Obviously, when the null hypothesis is correct, we want a low probability that hypothesis tests will produce statistically significant results. For example, if alpha is 0.05, your analysis has a 5% chance of producing a significant result when the null hypothesis is correct. Just as the evidentiary standard varies by the type of court case, you can set the significance level for a hypothesis test depending on the consequences of a false positive. By changing alpha, you increase or decrease the amount of evidence you require in the sample to conclude that the effect exists in the population. Learn more about Statistical Significance: Definition & Meaning. ## Changing Significance LevelsBecause 0.05 is the standard alpha, we’ll start by adjusting away from that value. Typically, you’ll need a good reason to change the significance level to something other than 0.05. Also, note the inverse relationship between alpha and the amount of required evidence. For instance, increasing the significance level from 0.05 to 0.10 lowers the evidentiary standard. Conversely, decreasing it from 0.05 to 0.01 increases the standard. Let’s look at why you would consider changing alpha and how it affects your hypothesis test. ## Increasing the Significance LevelImagine you’re testing the strength of party balloons. You’ll use the test results to determine which brand of balloons to buy. A false positive here leads you to buy balloons that are not stronger. The drawbacks of a false positive are very low. Consequently, you could consider lessening the amount of evidence required by changing the significance level to 0.10. Because this change decreases the amount of required evidence, it makes your test more sensitive to detecting differences, but it also increases the chance of a false positive from 5% to 10%. ## Decreasing the Significance LevelConversely, imagine you’re testing the strength of fabric for hot air balloons. A false positive here is very risky because lives are on the line! You want to be very confident that the material from one manufacturer is stronger than the other. In this case, you should increase the amount of evidence required by changing alpha to 0.01. Because this change increases the amount of required evidence, it makes your test less sensitive to detecting differences, but it decreases the chance of a false positive from 5% to 1%. It’s all about the tradeoff between sensitivity and false positives! In conclusion, a significance level of 0.05 is the most common. However, it’s the analyst’s responsibility to determine how much evidence to require for concluding that an effect exists. How problematic is a false positive? There is no single correct answer for all circumstances. Consequently, you need to choose the significance level! While the significance level indicates the amount of evidence that you require, the p-value represents the strength of the evidence that exists in your sample. When your p-value is less than or equal to the significance level, the strength of the sample evidence meets or exceeds your evidentiary standard for rejecting the null hypothesis and concluding that the effect exists. While this post looks at significance levels from a conceptual standpoint, learn about the significance level and p-values using a graphical representation of how hypothesis tests work. Additionally, my post about the types of errors in hypothesis testing takes a deeper look at both Type 1 and Type II errors, and the tradeoffs between them.
When you run an experiment or analyze data, you want to know if your findings are “significant.” But business relevance (i.e., practical significance) isn’t always the same thing as confidence that a result isn’t due purely to chance (i.e., statistical significance). This is an important distinction; unfortunately, statistical significance is often misunderstood and misused in organizations today. And yet because more and more companies are relying on data to make critical business decisions, it’s an essential concept for managers to understand. To better understand what statistical significance really means, I talked with Thomas Redman, author of Data Driven: Profiting from Your Most Important Business Asset. He also advises organizations on their data and data quality programs. ## What Is Statistical Significance?“Statistical significance helps quantify whether a result is likely due to chance or to some factor of interest,” says Redman. When a finding is significant, it simply means you can feel confident that’s it real, not that you just got lucky (or unlucky) in choosing the sample. When you run an experiment, conduct a survey, take a poll, or analyze a set of data, you’re taking a sample of some population of interest, not looking at every single data point that you possibly can. Consider the example of a marketing campaign. You’ve come up with a new concept and you want to see if it works better than your current one. You can’t show it to every single target customer, of course, so you choose a sample group. When you run the results, you find that those who saw the new campaign spent $10.17 on average, more than the $8.41 those who saw the old one spent. This $1.76 might seem like a big — and perhaps important — difference. But in reality you may have been unlucky, drawing a sample of people who do not represent the larger population; in fact, maybe there was no difference between the two campaigns and their influence on consumers’ purchasing behaviors. This is called a sampling error, something you must contend with in any test that does not include the entire population of interest. Redman notes that there are two main contributors to sampling error: the size of the sample and the variation in the underlying population. Sample size may be intuitive enough. Think about flipping a coin five times versus flipping it 500 times. The more times you flip, the less likely you’ll end up with a great majority of heads. The same is true of statistical significance: With bigger sample sizes, you’re less likely to get results that reflect randomness. All else being equal, you’ll feel more comfortable in the accuracy of the campaigns’ $1.76 difference if you showed the new one to 1,000 people rather than just 25. Of course, showing the campaign to more people costs more, so you have to balance the need for a larger sample size with your budget. Variation is a little trickier to understand, but Redman insists that developing a sense for it is critical for all managers who use data. Consider the images below. Each expresses a different possible distribution of customer purchases under Campaign A. In the chart on the left (with less variation), most people spend roughly the same amount of dollars. Some people spend a few dollars more or less, but if you pick a customer at random, chances are pretty good that they’ll be pretty close to the average. So it’s less likely that you’ll select a sample that looks vastly different from the total population, which means you can be relatively confident in your results. Compare that to the chart on the right (with more variation). Here, people vary more widely in how much they spend. The average is still the same, but quite a few people spend more or less. If you pick a customer at random, chances are higher that they are pretty far from the average. So if you select a sample from a more varied population, you can’t be as confident in your results. To summarize, the important thing to understand is that the greater the variation in the underlying population, the larger the sampling error.
See more HBR charts in Data & Visuals Redman advises that you should plot your data and make pictures like these when you analyze the data. The graphs will help you get a feel for variation, the sampling error, and, in turn, the statistical significance. No matter what you’re studying, the process for evaluating significance is the same. You start by stating a null hypothesis, often a straw man that you’re trying to disprove. In the above experiment about the marketing campaign, the null hypothesis might be “On average, customers don’t prefer our new campaign to the old one.” Before you begin, you should also state an alternative hypothesis, such as “On average, customers prefer the new one,” and a target significance level. The significance level is an expression of how rare your results are, under the assumption that the null hypothesis is true. It is usually expressed as a “p-value,” and the lower the p-value, the less likely the results are due purely to chance. Setting a target and interpreting p-values can be dauntingly complex. Redman says it depends a lot on what you are analyzing. “If you’re searching for the Higgs boson, you probably want an extremely low p-value, maybe 0.00001,” he says. “But if you’re testing for whether your new marketing concept is better or the new drill bits your engineer designed work faster than your existing bits, then you’re probably willing to take a higher value, maybe even as high as 0.25.” Note that in many business experiments, managers skip these two initial steps and don’t worry about significance until after the results are in. However, it’s good scientific practice to do these two things ahead of time. Then you collect your data, plot the results, and calculate statistics, including the p-value, which incorporates variation and the sample size. If you get a p-value lower than your target, then you reject the null hypothesis in favor of the alternative. Again, this means the probability is small that your results were due solely to chance. ## How is it calculated?As a manager, chances are you won’t ever calculate statistical significance yourself. “Most good statistical packages will report the significance along with the results,” says Redman. There is also a formula in Microsoft Excel and a number of other online tools that will calculate it for you. Still, it’s helpful to know the process described above in order to understand and interpret the results. As Redman advises, “Managers should not trust a model they don’t understand.” ## How Do Companies Use It?Companies use statistical significance to understand how strongly the results of an experiment, survey, or poll they’ve conducted should influence the decisions they make. For example, if a manager runs a pricing study to understand how best to price a new product, he will calculate the statistical significance — with the help of an analyst, most likely — so that he knows whether the findings should affect the final price. Remember that the new marketing campaign above produced a $1.76 boost (more than 20%) in average sales? It’s surely of practical significance. If the p-value comes in at 0.03 the result is also statistically significant, and you should adopt the new campaign. If the p-value comes in at 0.2 the result is not statistically significant, but since the boost is so large you’ll likely still proceed, though perhaps with a bit more caution. But what if the difference were only a few cents? If the p-value comes in at 0.2, you’ll stick with your current campaign or explore other options. But even if it had a significance level of 0.03, the result is likely real, though quite small. In this case, your decision probably will be based on other factors, such as the cost of implementing the new campaign.
Are You Using the Right Data to Power Your Digital Transformation? Closely related to the idea of a significance level is the notion of a confidence interval. Let’s take the example of a political poll. Say there are two candidates: A and B. The pollsters conduct an experiment with 1,000 “likely voters”; 49% of the sample say they’ll vote for A, and 51% say they’ll vote for B. The pollsters also report a margin of error of +/- 3%. “Technically,” says Redman, “49% +/-3% is a ‘95% confidence interval’ for the true proportion of A voters in the population.” Unfortunately, he says, most people interpret this as “there’s a 95% chance that A’s true percentage lies between 46% and 52%,” but that isn’t correct. Instead, it says that if the pollsters were to do the result many times, 95% of intervals constructed this way would contain the true proportion. If your head is spinning at that last sentence, you’re not alone. As Redman says, this interpretation is “maddeningly subtle, too subtle for most managers and even many researchers with advanced degrees.” He says the more practical interpretation of this would be “Don’t get too excited that B has a lock on the election” or “B appears to have a lead, but it’s not a statistically significant one.” Of course, the practical interpretation would be very different if 70% of the likely voters said they’d vote for B and the margin of error was 3%. The reason managers bother with statistical significance is they want to know what findings say about what they should do in the real world. But “confidence intervals and hypothesis tests were designed to support ‘science,’ where the idea is to learn something that will stand the test of time,” says Redman. Even if a finding isn’t statistically significant, it may have utility to you and your company. On the other hand, when you’re working with large data sets, it’s possible to obtain results that are statistically significant but practically meaningless, like that a group of customers is 0.000001% more likely to click on Campaign A over Campaign B. So rather than obsessing about whether your findings are precisely right, think about the implication of each finding for the decision you’re hoping to make. What would you do differently if the finding were different? ## What Mistakes Do People Make When Working with Statistical Significance?“Statistical significance is a slippery concept and is often misunderstood,” warns Redman. “I don’t run into very many situations where managers need to understand it deeply, but they need to know how to not misuse it.” Of course, data scientists don’t have a monopoly on the word “significant,” and often in businesses it’s used to mean whether a finding is strategically important. It’s good practice to use language that’s as clear as possible when talking about data findings. If you want to discuss whether the finding has implications for your strategy or decisions, it’s fine to use the word “significant,” but if you want to know whether something is statistically significant (and you should want to know that), be precise in your language. Next time you look at results of a survey or experiment, ask about the statistical significance if the analyst hasn’t reported it.
A Refresher on Regression Analysis Remember that statistical significance tests help you account for potential sampling errors, but Redman says what is often more worrisome is the non-sampling error: “Non-sampling error involves things where the experimental and/or measurement protocols didn’t happen according to plan, such as people lying on the survey, data getting lost, or mistakes being made in the analysis.” This is where Redman sees more-troubling results. “There is so much that can happen from the time you plan the survey or experiment to the time you get the results. I’m more worried about whether the raw data is trustworthy than how many people they talked to,” he says. Clean data and careful analysis are more important than statistical significance. Always keep in mind the practical application of the finding. And don’t get too hung up on setting a strict confidence interval. Redman says there’s a bias in scientific literature that “a result wasn’t publishable unless it hit a p = 0.05 (or less).” But for many decisions — like which marketing approach to use — you’ll need a much lower confidence interval. In business, Redman says, there’s often more-important criteria than statistical significance. The important question is, “Does the result stand up in the market, if only for a brief period of time?” As Redman says, the results only give you so much information: “I’m all for using statistics, but always wed it with good judgment.” |