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Therapists regularly perform various measurements. How reliable these measurements are in themselves, and how reliable therapists are in using them, is clearly essential knowledge to help clinicians decide whether or not a particular measurement is of any value. The aim of this paper is to explain the nature of reliability, and to describe some of the commonly used estimates that attempt to quantify it. An understanding of reliability, and how it is estimated, will help therapists to make sense of their own clinical findings, and to interpret published studies.
Although reliability is generally perceived as desirable, there is no firm definition as to the level of reliability required to reach clinical acceptability. As with hypothesis testing, statistically significant levels of reliability may not translate into clinically acceptable levels, so that some authors' claims about reliability may need to be interpreted with caution. Reliability is generally population specific, so that caution is also advised in making comparisons between studies.
DEFINITION OF RELIABILTY
Reliability is an engineering discipline for applying scientific know-how to a component, assembly, plant, or process so it will perform its intended function, without failure, for the required time duration when installed and operated correctly in a specified environment.
Reliability terminates with a failure-i.e, unreliability occurs. Business enterprises observe the high cost of unreliability. The high cost of unreliability motivates an engineering solution to control and reduce costs.
Among living organisms, reliability would be studied in terms of survivors. Unreliability would be studied in terms of mortality.
You may not clearly understand the definition of reliability. If your automobile stops functioning during your mission, you will clearly understand the concept of unreliability. You'll also learn about the gut rending reality of the cost of unreliability when you have your automobile restored to a reliable condition.
MIL-STD-721C and MIL-HDBK-338 have Definitions of Terms For Reliability and Maintainability and they give two definitions for reliability:
The duration or probability of failure-free performance under stated conditions
The probability than an item can perform its intended function for a specified interval under stated conditions
(For non-redundant items this is equivalent to definition (1).
For redundant items this is equivalent to definition of mission reliability)
Reliability is the probability that a device, system, or process will perform its prescribed duty without failure for a given time when operated correctly in a specified environment.
Reliability engineering is an engineering field, that deals with the study of reliability: the ability of a system or component to perform its required functions under stated conditions for a specified period of time. It is often reported as a probability.
Reliability may be defined in several ways:
The idea that something is fit for purpose with respect to time;
The capacity of a device or system to perform as designed;
The resistance to failure of a device or system;
The ability of a device or system to perform a required function under stated conditions for a specified period of time;
The probability that a functional unit will perform its required function for a specified interval under stated conditions.
The ability of something to "fail well" (fail without catastrophic consequences)
Reliability engineers rely heavily on statistics, probability theory, and reliability theory. Many engineering techniques are used in reliability engineering, such as reliability prediction, Weibull analysis, thermal management, reliability testing and accelerated life testing. Because of the large number of reliability techniques, their expense, and the varying degrees of reliability required for different situations, most projects develop a reliability program plan to specify the reliability tasks that will be performed for that specific system.
The function of reliability engineering is to develop the reliability requirements for the product, establish an adequate reliability program, and perform appropriate analyses and tasks to ensure the product will meet its requirements. These tasks are managed by a reliability engineer, who usually holds an accredited engineering degree and has additional reliability-specific education and training. Reliability engineering is closely associated with maintainability engineering and logistics engineering. Many problems from other fields, such as security engineering, can also be approached using reliability engineering techniques. This article provides an overview of some of the most common reliability engineering tasks. Please see the references for a more comprehensive treatment
In this context the definition of reliability is straightforward: a measurement is reliable if it reflects mostly true score, relative to the error. For example, an item such as "Red foreign cars are particularly ugly" would likely provide an unreliable measurement of prejudices against foreign- made cars. This is because there probably are ample individual differences concerning the likes and dislikes of colors. Thus, this item would "capture" not only a person's prejudice but also his or her color preference. Therefore, the proportion of true score (for prejudice) in subjects' response to that item would be relatively small.
Measures of reliability. From the above discussion, one can easily infer a measure or statistic to describe the reliability of an item or scale. Specifically, we may define an index of reliability in terms of the proportion of true score variability that is captured across subjects or respondents, relative to the total observed variability. In equation form, we can say:
What will happen when we sum up several more or less reliable items designed to measure prejudice against foreign-made cars? Suppose the items were written so as to cover a wide range of possible prejudices against foreign-made cars. If the error component in subjects' responses to each question is truly random, then we may expect that the different components will cancel each other out across items. In slightly more technical terms, the expected value or mean of the error component across items will be zero. The true score component remains the same when summing across items. Therefore, the more items are added, the more true score (relative to the error score) will be reflected in the sum scale.
Number of items and reliability. This conclusion describes a basic principle of test design. Namely, the more items there are in a scale designed to measure a particular concept, the more reliable will the measurement (sum scale) be. Perhaps a somewhat more practical example will further clarify this point. Suppose you want to measure the height of 10 persons, using only a crude stick as the measurement device. Note that we are not interested in this example in the absolute correctness of measurement (i.e., in inches or centimeters), but rather in the ability to distinguish reliably between the 10 individuals in terms of their height. If you measure each person only once in terms of multiples of lengths of your crude measurement stick, the resultant measurement may not be very reliable. However, if you measure each person 100 times, and then take the average of those 100 measurements as the summary of the respective person's height, then you will be able to make very precise and reliable distinctions between people (based solely on the crude measurement stick).
An alternative way of computing the reliability of a sum scale is to divide it in some random manner into two halves. If the sum scale is perfectly reliable, we would expect that the two halves are perfectly correlated (i.e., r = 1.0). Less than perfect reliability will lead to less than perfect correlations. We can estimate the reliability of the sum scale via the Spearman-Brown split half coefficient:
rsb = 2rxy /(1+rxy)
In this formula, rsb is the split-half reliability coefficient, and rxy represents the correlation between the two halves of the scale.
Correction for Attenuation
Let us now consider some of the consequences of less than perfect reliability. Suppose we use our scale of prejudice against foreign-made cars to predict some other criterion, such as subsequent actual purchase of a car. If our scale correlates with such a criterion, it would raise our confidence in the validity of the scale, that is, that it really measures prejudices against foreign-made cars, and not something completely different. In actual test design, the validation of a scale is a lengthy process that requires the researcher to correlate the scale with various external criteria that, in theory, should be related to the concept that is supposedly being measured by the scale.
How will validity be affected by less than perfect scale reliability? The random error portion of the scale is unlikely to correlate with some external criterion. Therefore, if the proportion of true score in a scale is only 60% (that is, the reliability is only .60), then the correlation between the scale and the criterion variable will be attenuated, that is, it will be smaller than the actual correlation of true scores. In fact, the validity of a scale is always limited by its reliability.
Given the reliability of the two measures in a correlation (i.e., the scale and the criterion variable), we can estimate the actual correlation of true scores in both measures. Put another way, we can correct the correlation for attenuation:
rxy,corrected = rxy /(rxx*ryy)Â½
In this formula, rxy,corrected stands for the corrected correlation coefficient, that is, it is the estimate of the correlation between the true scores in the two measures x and y. The term rxy denotes the uncorrected correlation, and rxx and ryy denote the reliability of measures (scales) x and y. You can compute the attenuation correction based on specific values, or based on actual raw data (in which case the reliabilities of the two measures are estimated from the data).
Designing a Reliable Scale
After the discussion so far, it should be clear that, the more reliable a scale, the better (e.g., more valid) the scale. As mentioned earlier, one way to make a sum scale more valid is by adding items. You can compute how many items would have to be added in order to achieve a particular reliability, or how reliable the scale would be if a certain number of items were added. However, in practice, the number of items on a questionnaire is usually limited by various other factors (e.g., respondents get tired, overall space is limited, etc.). Let us return to our prejudice example, and outline the steps that one would generally follow in order to design the scale so that it will be reliable:
Step 1: Generating items. The first step is to write the items. This is essentially a creative process where the researcher makes up as many items as possible that seem to relate to prejudices against foreign-made cars. In theory, one should "sample items" from the domain defined by the concept. In practice, for example in marketing research, focus groups are often utilized to illuminate as many aspects of the concept as possible. For example, we could ask a small group of highly committed American car buyers to express their general thoughts and feelings about foreign-made cars. In educational and psychological testing, one commonly looks at other similar questionnaires at this stage of the scale design, again, in order to gain as wide a perspective on the concept as possible.
Step 2: Choosing items of optimum difficulty. In the first draft of our prejudice questionnaire, we will include as many items as possible. We then administer this questionnaire to an initial sample of typical respondents, and examine the results for each item. First, we would look at various characteristics of the items, for example, in order to identify floor or ceiling effects. If all respondents agree or disagree with an item, then it obviously does not help us discriminate between respondents, and thus, it is useless for the design of a reliable scale. In test construction, the proportion of respondents who agree or disagree with an item, or who answer a test item correctly, is often referred to as the item difficulty. In essence, we would look at the item means and standard deviations and eliminate those items that show extreme means, and zero or nearly zero variances.
Step 3: Choosing internally consistent items. Remember that a reliable scale is made up of items that proportionately measure mostly true score; in our example, we would like to select items that measure mostly prejudice against foreign-made cars, and few esoteric aspects we consider random error. To do so, we would look at the following:
Shown above are the results for 10 items. Of most interest to us are the three right-most columns. They show us the correlation between the respective item and the total sum score (without the respective item), the squared multiple correlation between the respective item and all others, and the internal consistency of the scale (coefficient alpha) if the respective item would be deleted. Clearly, items 5 and 6 "stick out," in that they are not consistent with the rest of the scale. Their correlations with the sum scale are .05 and .12, respectively, while all other items correlate at .45 or better. In the right-most column, we can see that the reliability of the scale would be about .82 if either of the two items were to be deleted. Thus, we would probably delete the two items from this scale.
Step 4: Returning to Step 1. After deleting all items that are not consistent with the scale, we may not be left with enough items to make up an overall reliable scale (remember that, the fewer items, the less reliable the scale). In practice, one often goes through several rounds of generating items and eliminating items, until one arrives at a final set that makes up a reliable scale.
Â Reliability theory
Reliability theory is the foundation of reliability engineering. For engineering purposes, reliability is defined as:
the probability that a device will perform its intended function during a specified period of time under stated conditions.
Mathematically, this may be expressed as,
where is the failure probability density function and t is the length of the period of time (which is assumed to start from time zero).
Reliability engineering is concerned with four key elements of this definition:
First, reliability is a probability. This means that failure is regarded as a random phenomenon: it is a recurring event, and we do not express any information on individual failures, the causes of failures, or relationships between failures, except that the likelihood for failures to occur varies over time according to the given probability function. Reliability engineering is concerned with meeting the specified probability of success, at a specified statistical confidence level.
Second, reliability is predicated on "intended function:" Generally, this is taken to mean operation without failure. However, even if no individual part of the system fails, but the system as a whole does not do what was intended, then it is still charged against the system reliability. The system requirements specification is the criterion against which reliability is measured.
Third, reliability applies to a specified period of time. In practical terms, this means that a system has a specified chance that it will operate without failure before time . Reliability engineering ensures that components and materials will meet the requirements during the specified time. Units other than time may sometimes be used. The automotive industry might specify reliability in terms of miles, the military might specify reliability of a gun for a certain number of rounds fired. A piece of mechanical equipment may have a reliability rating value in terms of cycles of use.
Fourth, reliability is restricted to operation under stated conditions. This constraint is necessary because it is impossible to design a system for unlimited conditions. A Mars Rover will have different specified conditions than the family car. The operating environment must be addressed during design and testing. Also, that same rover, may be required to operate in varying conditions requiring additional scrutiny.
Design for reliability
Design For Reliability (DFR), is an emerging discipline that refers to the process of designing reliability into products. This process encompasses several tools and practices and describes the order of their deployment that an organization needs to have in place to drive reliability into their products. Typically, the first step in the DFR process is to set the system's reliability requirements. Reliability must be "designed in" to the system. During system design, the top-level reliability requirements are then allocated to subsystems by design engineers and reliability engineers working together.
Reliability design begins with the development of a model. Reliability models use block diagrams and fault trees to provide a graphical means of evaluating the relationships between different parts of the system. These models incorporate predictions based on parts-count failure rates taken from historical data. While the predictions are often not accurate in an absolute sense, they are valuable to assess relative differences in design alternatives.
A Fault Tree Diagram
One of the most important design techniques is redundancy. This means that if one part of the system fails, there is an alternate success path, such as a backup system. An automobile brake light might use two light bulbs. If one bulb fails, the brake light still operates using the other bulb. Redundancy significantly increases system reliability, and is often the only viable means of doing so. However, redundancy is difficult and expensive, and is therefore limited to critical parts of the system. Another design technique, physics of failure, relies on understanding the physical processes of stress, strength and failure at a very detailed level. Then the material or component can be re-designed to reduce the probability of failure. Another common design technique is component derating: Selecting components whose tolerance significantly exceeds the expected stress, as using a heavier gauge wire that exceeds the normal specification for the expected electrical current.
Results are presented during the system design reviews and logistics reviews. Reliability is just one requirement among many system requirements. Engineering trade studies are used to determine the optimum balance between reliability and other requirements and constraints.
A Reliability Sequential Test Plan
The purpose of reliability testing is to discover potential problems with the design as early as possible and, ultimately, provide confidence that the system meets its reliability requirements.
Reliability testing may be performed at several levels. Complex systems may be tested at component, circuit board, unit, assembly, subsystem and system levels. (The test level nomenclature varies among applications.) For example, performing environmental stress screening tests at lower levels, such as piece parts or small assemblies, catches problems before they cause failures at higher levels. Testing proceeds during each level of integration through full-up system testing, developmental testing, and operational testing, thereby reducing program risk. System reliability is calculated at each test level. Reliability growth techniques and failure reporting, analysis and corrective active systems (FRACAS) are often employed to improve reliability as testing progresses. The drawbacks to such extensive testing are time and expense. Customers may choose to accept more risk by eliminating some or all lower levels of testing.
It is not always feasible to test all system requirements. Some systems are prohibitively expensive to test; some failure modes may take years to observe; some complex interactions result in a huge number of possible test cases; and some tests require the use of limited test ranges or other resources. In such cases, different approaches to testing can be used, such as accelerated life testing, design of experiments, and simulations.