# Introduction

In this section, you will learn about constructing Time-between charts and interpreting the findings.  We focus on constructing a control chart for your diet and exercise patterns.  In constructing control chart for diet or exercise, we assume that you have made a change in your life style (daily processes and routines) and have collected data and are wondering if the change has led to improvement.  Since your weight and exercise time vary for many reasons, the key question is whether current weight and exercise time is better than historical patterns.

This section assumes that you can plot data, take a square root and calculate means.  These are relatively simple tasks but some people may have little experience with any data manipulation.  Tutorials on how to do these tasks are also available at end of this section.

# Why Construct a Control Chart

A control chart is constructed to help guide our intuitions.  Most people read too much into their success and attribute their failures to external events.  Control charts can help discipline intuitions about success and failure.  For example, in weight loss there are considerable variations depending on timing of weight measurement,  instruments used to measure the weight, clothes on the person while weight is measured, recent food intake, whether, and many other sources.  These variations lead to unreliability in the measure of weight.  It would be a fallacy to see these variations as weight loss or gain.  Control charts can help remove the guess work.  These charts establish if new values are different from historical values.  Control charts can help answer whether your new weight and exercise patterns indicate a departure from their historical levels.

# What is a Control Chart?

In a control chart, you monitor your progress over time.  You create a plot, where the X-axis is days since start and the Y-axis is the outcome you are monitoring.  To decide if your outcomes are different from historical patterns, the upper (UCL) and lower control limits (LCL) are calculated.  These limits are organized in such a way as to make sure that if your historical pattern has continued then 99% of time data will fall within these limits.  The upper and lower control limits are calculated using mathematical formulas that are specific to the type of outcome you are monitoring.  This section shows you how to calculate these limits depending on whether you are monitoring your weight, your exercise time, days diet missed, days exercise missed, or other similar outcomes.

Figure 1 shows the structure of a typical control chart.  In this figure, all points, except for one, fall within the control limits.

Figure 1 shows the structure of a typical control chart.

# How to Read a Control Chart?

A control chart is useful in many different ways.  Points outside the limits are unusual and mark departure from historical patterns.  You have lost weight if your new measure is below the lower control limit.  Two points in Figure 1 fall below the LCL and therefore mark a weight loss.  All other points do not indicate any real weight loss, even though there are lots of them showing a decrease in weight.  These small fluctuations are random and not different from your historical changes rise and falls in your weight.

In Figure 1, none of the points fall above the upper control limits; therefore the person has not gained weight.

You can also use the control chart to see if you are maintaining your gains in a previous time periods.  If your data falls within the control limits, despite day to day variations, there has not been any change in your weight and exercise.   If you are at ideal weight and exercise, then you want your data to fall within the limits.

# Minimum Number of Observations

The more data you have, the more precision you have in constructing the upper and lower control limits.   Not all of the data are used for calculation of control limits.  Often, the limits are based on pre-intervention period.  Then subsequent post-intervention observations are compared to the pre-intervention limits.  At a minimum, you need at least 7 data points in the pre-intervention period to start most charts.  When you make a change, you want to see if your weight and exercise have been affected by the change.  In these circumstances, you set the limits based on the pre-intervention data.  You compare post-intervention findings to these limits.  If any points fall outside the limits, you can then conclude that the intervention has changed your weight or exercise patterns.  See Figure 2 for an example of limits set based on pre-intervention periods.

Figure 2:  An example of limits set based on pre-intervention periods

Compare the chart in Figure 2 with the chart in Figure 1.  Both are based on the same data, but in Figure 2 the limits are based on the first 7 days, before the intervention.  Figure 2 shows that post intervention data are lower than LCL and therefore a significant change has occurred.  When Figure 2 is compared to Figure 1, we see that more points are out of the limits in Figure 2.  By setting the limits to pre-intervention patterns, we were able to detect more accurately the improvements since the intervention.

The length of data used in construction of control limit depends on the timing of the intervention and changes in the underlying process. Use about 15 data points before the start of the intervention to set the control limit. You can of course use more data points to get a more stable picture of the process but keep in mind that as you use more data points you are going back further in time.  The more distant the data the less relevant it is to the current situation.  There is a practical limit of how far back can you go to collect the data you need.  Taking data from months ago may make your analysis less accurate if the process has changed since then.

# Assumptions of Time-between Charts

 Figure 3:  Different Charts for Different Data Time-between chart is best when there are single observations per period, outcomes are dichotomous (not interval scale) and the event of interest is rare

Time-between charts is one method of constructing control charts, there are many more ways to construct a control chart.  Time between charts are best suited when four assumptions are met:

1.  Data should have been collected over time with one observation per time period.
2. The chart should be drawn for dichotomous, discrete rare event.   For example, Time-between charts can be constructed for days diet-missed, days exercise-missed, days without coffee, days without junk food, etc.
3. Observations over time should be independent of each other.  Knowing the value of observation at one time period should not change the probability of observation at next time period.
4. The time to the event should have a Geometric distribution, in which longer time to the event is increasingly more rare.

There are many other methods of control chart that are also available (see Figure 3 for examples).   You could use a Tukey chart for analysis of continuous data such as length of exercise time or weight.

# Calculating Limits for Time-between Charts

The steps in constructing control limits for time in between charts are:

1.  Verify that days missed are fewer than days in which you kept up with the plan.  Control limits must be derived on rare events.  If day missed is more common, it is important to plot days the person kept up with the plan.  If days the person kept up with the plan, it is important to chart days missed.

2. Calculate consecutive time between events.  Consecutive days between events are calculated based on what happened in the previous day and today.  Table 1 shows how these values are calculated for length of missed days and length of days habit kept. The analysis either plots consecutive missed-days or consecutive days habit kept based on which one happens less often.  If missed days are less often, the chart is constructed by plotting length of missed-days on the Y-axis and time since start on the X-axis.  If otherwise, the control chart is constructed by calculating the consecutive days in which the habit was kept.

 Yesterday Today Length of missed-days Length of days habit kept No data Missed day 1 day 0 day No data Habit kept 0 day 1 day Habit kept Habit kept 0 day 1 + yesterday's length of days habit kept Missed day Habit kept 0 day 1 day Habit kept Missed day 1 day 0 day Missed day Missed day 1 + yesterday’s length of Missed day 0 day Table 1:  Rules for Calculating length of time between missed-days
1. Calculate R, the ratio of days missed to days keeping up with plan:

The value of R muse always be less than one.   If the data distinguish between pre- and post-intervention periods, then the value of R is calculated either based on pre- or post-intervention data.  The period used for calculation of R is selected so that it would minimize the value of R.  If the pre-intervention period has little variation in the duration of the event being tracked, then R is calculated from the pre-intervention period.  Otherwise, it is calculated from the post intervention period.
Then, comparing the observed data to the control limits allows us to examine the impact of the intervention.

1. Calculate UCL as R plus 3 times the square root of R times one plus R.

There is no Lower Control Limit (LCL) for Time-between Charts.  As the event plotted is rare, the LCL will always be  a negative number.  Since time cannot be negative, the LCL does not make sense in the context of Time-between Charts.

2. Plot either duration of days missed or days kept up with the plan against time.     Check to see if the duration exceeds UCL.

Please note that time between charts are about the a series of consecutive events and not about a specific point in the series.  When you look at a chart for failure, for example, and see a string of consecutive failures, and one point of this string is above the upper control limit, the interpretation is that the entire series is unusual and not just the point that is above the control limit. Strictly speaking, time in between charts for failures should be drawn as sum of continuous days of failure. This means that the chart would be on 0 for days of success and when a string of failures start, would have no value until the end of the string, at which point it will the value will be at the sum of days in the string. This produces a chart with lots of discontinuous events. To make the interpretation of the chart easier we draw the days of failure from the start of the string till its end. So we draw the first day, the second day until the last day in the string of failures. This gives a more continuous feeling to the chart. Even though we have changed how the chart looks, the statistical tests are still done as before on the end point of the series.

# An Example

Table 2 shows data collected over 18 days by a 35 year old female trying to exercise more.  She decided to take morning showers at the gym and thus combine her exercise and shower routines.  The first 10 days show the data before the intervention.  The remaining days show the data after the intervention.  The question was whether this new habit has led to increased use of the gym.

 Day Missed? Duration of missed-days 1 No 0 2 Yes 1 3 Yes 2 4 Yes 3 5 No 0 6 Yes 1 7 Yes 2 8 No 0 9 No 0 10 No 0 11 No 0 12 No 0 13 No 0 14 No 0 15 No 0 16 Yes 1 17 No 0 18 No 0 R =0.13 Table 2: Missed-days of exercise

To construct the control chart, we first need to use the rules in Table 1 to calculate the duration of missed-days in Table 2.  Note that missed-days grow in length until she goes to the gym, at which point they are re-set to zero.  The last column in Table 2 shows the calculated length of missed-days.  The control limit can be calculated from either the pre- or the post-intervention data, which ever leads to a lower upper control limit.  In this case the control limit is calculated form the post intervention data, the data for days 8 through 18, because it has the least variability.  There is 1 missed day and 8 days she has kept up with plans.  Therefore, R is calculated as 1/8 = 0.13.  The UCL is then calculated as:

Figure below shows the resulting chart and control limit.

Figure 4:  Analysis of Data in Table 2

# Interpretation of Time-between Control Chart

If the observations in the control chart exceed the Upper Control Limit, then these observations are unlikely to occur by chance.  They signify a change in the underlying frequency of the event being tracked.  If the control limits were based on pre- or post-intervention periods, observations above control limit indicate the impact of the intervention.  Of course, it is possible that the change in probability of the event might be due to another event not tracked in the control chart.  Therefore, attribution of change in the probability of the event to the intervention should be made with caution.

The chart in Figure 5 shows that in the pre-intervention period the patient had two strings of missed-days.  In the first string, she did not go to the gym for 4 days.  In the second string, she did not go for 2 consecutive days.  Both strings exceed the UCL.  Compared to post intervention period, these two strings of missed-days are long enough to constitute a real change in the process.  Based on these findings, we conclude that the intervention was working and the rate of missed-days has dropped.  It is however possible that the rate of missed-days dropped for another reason besides taking showers at the gym.

# Conclusion

The point of any control chart is to help you improve.  The effort we put into measurement and analysis is wasted if it does not help us improve.  Constructing a control chart is time consuming and for some difficult.  But what is the alternative.  Many err in detecting real changes in their weight and exercise times.  They mistake random fluctuations for real progress.  Control charts help discipline our intuitions to see beyond the rise and fall of weight and exercise patterns.

# Presentations

There are six sets of presentations for this lecture:

1. Time-between charts for missed exercise Slides Listen Video►  SWF

2. How to plot a Control Chart  Slides  Listen

3. Introduction to Control Chart  Slides  Listen

5. Time-between  charts in asthma care.  The narrated slides include formulas used in Excel to calculate control charts.  These formulas are not as easy to see in the Video.  Slides  Listen Video  SWF

6. See video on calculation of UCL based on post intervention period for Time-between chart in Excel.  Note that in this video the formula for Countif function in Excel differs from previous videos.  Both set are correct but calculated in different ways.  Video

Listening to narrated slides and videos may require Flash

# Assignment Due this Week

1. Every week ask a question or comment on the lecture.  Comment►  Ask►

2. Using Excel, analyze the data in Table 2 on this web page.  Except for the data in the table, all other entries should be calculated cell values and not entered cell values.

3. Report how many data points you have collected in your personal improvements and whether the data you have meets the assumptions of Time between Control Charts.

Email your work to the instructor.  Make sure that you include the course number in the subject line.  Include the your course-specific student ID number in the subject line.  Make sure that you attach your Excel file.  Put answer to question 3 in the email.

# More

• Time-between chart for Asthma care.  Alemi and  colleagues provide an example of how  patients can use Time-between charts to analyze diaries of asthmatic patients.    PubMed►

• Alemi and colleagues describe how judges in Family Drug Court can use Time-between charts to help them decide if patients' relapse is a return to old patterns.

• Benneyan JC. Number-between g-type statistical quality control charts for monitoring adverse events. Health Care Management Science. 2001 Dec; 4(4): 305-18.  More►

• Benneyan JC. Performance of number-between g-type statistical control charts for monitoring adverse events.  Health Care Management Science. 2001 Dec; 4(4): 319-36.  More►

This page is copyright protected by Farrokh Alemi, Ph.D..  This page is part of the  course on lifestyle management. This page was first made on  Wednesday, November 06, 2002 and most recent revision was on 1/1/2006