Time measurements differ from other measurements. All measurements of time are positive values above zero, and the variance of time increases with its value. Time therefore conforms to the exponential distribution.

Where the range of time measurement is narrow and far from the zero value, it can be considered as approximately normally distributed. An example is maternal age, ranging from the late teens to the mid forties. In most research reports, maternal age is handled as if it is a normally distributed measurement.

Where the range of time measurements is wide and encroaching towards 1, the variance required need to be stabilized. Often this can be achieved. Gestational age is such a measurement that needs logarithmic transformation for it to be handled as a continuous measurement.

To handle time measurements precisely, it should be handled as exponentially distributed. This requires complex and inflexible algorithms that limit its use, and usually some form of curvilinear transformation is carried out before it is handled as a normally distributed measurement. A transformation designed to do this is the Box Cox transformation, which is provided in the Numerical Transformation Program Page .

Up to this point, time is discussed on the basis that it can be easily and uniformly obtained, but this is not the case in much of medical research. Often the time is measured over a prolong period, perhaps years. Subjects are recruited along the way, and some will be lost before the event of interest occurs. At any time the data being analysed, the subjects will be varied in the time they have been in the study, and the event of interest may have occurred only in a [proportion of the subjects.

Survival Analysis is a term used to analyse time to events under these difficult situations. The data is said to be Censored, in the sense that, at the time of analysis, not all the required information is available, and the statistical procedures make do with what is available.

The are many complex and precise method of survival analysis. StatTools presents only the simplest and the most commonly used one, The Log Rank Test, as described by Kaplan Meier, and made popular by Peto (see references).

The Log Rank Test is used to evaluate time related change in proportions of an indexed event. Medically, it most commonly refer to death rate in cancer patients, such as the 5 year survival rate. However, the methodology has much wider use, such as time related recurrence rate, cure rate, discharge rate, pregnancy rate. In industry and manufacturing, it is also used to evaluate the lifetime of products such as how long before a light bulb burns outs.

The terminology can be a bit confusing at time. In cancer treatment, the survival rate is used, indicating the proportion of patients that are still alive at a fixed time after entering the study. However, death rate is also used, and the term hazard refers to the opposite of survival.

Mathematically, death and survival are two sides of the same coin, and the comparison of survival or death use the same statistical methods.

The Log Rank Test is particularly useful in cancer cases, as patients enter the study at different times, and because the follow up is usually in years, some are lost to follow up. At the time of analysis therefore, some may have died or are lost to follow up after varying intervals, and some may have joined the study recently, less than the full period of assessment. The Log Rank Test collates the available data and produces a survival rate that can be statistically compared.

Terminology

An interval is a unit of time. In cancer follow up this is usually a year, in light bulb burnt out it is often an hour, in IVF it is often a cycle, in contraception and pregnancy it is often a month.

An indexed event is the event of interest. In cancer this is usually death or recurrence, in light bulb it is burnt out, in IVF or contraception this is pregnancy.

The survival rate is the proportion of the subjects that have not as yet experienced the indexed event, and the hazard rate is the proportion that have experienced the indexed event. (survival rate = 1 - hazard rate).

An event table is a table that display the events in time. The rows are the intervals, and at each row each group of subjects has 2 columns, the number of subjects that experienced the indexed event (died) in that interval, and the number that survived that interval without the indexed event. As the interval proceed, the number of subjects decrease, some because they have experienced the indexed event (died), some because they are lost to follow up, some because they only recently join the study and have not been in the follow up long enough.

Please Note : The event table as described in the last paragraph applies to programs and explanation provided by StatTools, in common with many other presentations. However, users should take care, as another convention is to present the two columns as the total number of cases that entered the interval, and the number of cases that died during that interval. It is quite easy to convert one format to the other, but it is important to know the difference exists, so as not to get confused.

The rates are statistically compared after a fixed number of intervals. For example, 5 years survival for cancer patients, 100 hour not burnt out in light bulbs.

Format of Data Entry : The program in the Survival - Kaplan Meier Log Rank Test Program Page allows two format for data entry.

• The first is the most convenient for users. The data is a matrix with 3 columns, and data from each subject is in a row. Col 1 designates group (1,2,3, and so on). Col 2 is the number of intervals the subject has been in the study at the time of analysis. Col 3 is the fate of the subject. Zero (0) indicates that the subject has experienced the indexed event (died) at that interval. Any other value is taken that the subject has survived at the end of that period.

  The program converts this primary data into an event table, and performs the analysis from the event table.

• The second format is to enter the event table itself, if this is already compiled. The table has as many rows as the intervals. Each group has two columns, the first containing the number of of subjects that experienced the indexed event at the various intervals, and the other the number of subjects that is known to have survived that interval.

Sample size and power considerations is based on that for Log Rank Test in the text book by Pinol et.al. (see references), The calculations are presented in the Sample Size for Survival (Kaplan Meier Log Rank Test) Program Page and table of sample sizes in Sample Size for Survival (Kaplan Meier Log Rank Test) Explained and Tables Page .

Calculating the sample size requires 5 parameters.

• Alpha (α), the probability of Type I error.
• Power (1 - β)
• the anticipated proportions (either survival or hazard) in the two groups
• The ratio of numbers in the two groups. This is needed, particularly in cancer cases, as the number of subjects available in different groups often vary widely.
• As the group sizes may be different, the sample size required is that for two groups, to be divided up according to the ratio.
• When there are more than two groups, sample size should be calculated as a series of two group studies, without the need for Bonferroni's correction.
• The result is the total sample size (group 1 and 2 combined). In a two group study, where the sample size is the same, the sample size per group is half of that presented. Sample size for both One and Two tail studies are presented.
  • In many two group studies, particularly for different treatments in cancer, the interest is whether the new treatment is better than the old. In these cases the one tail model is appropriate, as it is more powerful
  • However, if the overall outcome is the interest, whether mortality is decreased by a more potent treatment but increase because of greater toxicity, then the two tail model is appropriate.
  • For multiple group studies, the main interest is whether the groups have different outcomes, regardless of directions, so the two tail model is more appropriate.

Power calculation also requires 5 parameters

• Alpha (α), the probability of Type I error.
• The sample size obtained in group 1
• The survival rate of group 1
• The sample size obtained in group 2
• The survival rate of group 2
• When there are more than two groups, power should be calculated as a series of two group studies, without the need for Bonferroni's correction.

This section uses the default example data from Survival - Kaplan Meier Log Rank Test Program Page to demonstrate the Log Rank Test. The data is computer generated to demonstrate the procedure, and does not reflect real information.

Tmt	Months	Pregnant
2	4	no
1	21	yes
2	15	no
2	1	no
2	6	no
1	10	no
1	19	yes
2	4	no
1	20	yes
2	20	yes
1	22	yes
2	4	no
1	5	no
2	10	no
2	4	no
2	3	no
2	24	yes
2	6	no
1	9	no
2	13	no
1	19	yes
1	7	no
1	14	no
2	21	yes
1	12	no
1	16	yes
1	9	no
1	13	no
2	15	no
1	23	yes
2	8	no
2	16	no
2	13	no
1	22	yes
2	12	no
2	24	yes
1	16	yes
2	8	no
1	18	yes
2	19	no

We are treating a group of women who are diagnosed to have polycystic ovarian disease, and are infertile. We are testing two protocols of ovarian stimulation, and comparing their effectiveness. We have decided that the difference in effectiveness should be that of pregnancy rate over 2 years (24 months).

	Tmt 1		Tmt 2
Intv	Pregnant	Not Pregnant	Pregnant	Not Pregnant
0	0	18	0	22
1	0	18	1	21
2	0	18	0	21
3	0	18	1	20
4	0	18	4	16
5	1	17	0	16
6	0	17	2	14
7	1	16	0	14
8	0	16	2	12
9	2	14	0	12
10	1	13	1	11
11	0	13	0	11
12	1	12	1	10
13	1	11	2	8
14	1	10	0	8
15	0	10	2	6
16	0	10	1	5
17	0	8	0	5
18	0	8	0	5
19	0	7	1	4
20-24	0	5	0	4

As this is a clinical situation, we do not have a cohort that starts at the same time. Over two years, we recruit those women that satisfy our criteria into our trial. At the end of two years, we have 24 women in the study, with the data as shown in the table to the right.

Having replaced "yes" with 0 and "no" with 1, the data was entered for analysis, and the event table is as presented to the left.

From this table, it can be seen that those receiving treatment 2 had ealier and more frequent successes in terms of pregnancy.

The survival curve is as shown in the plot to the right. Here survival means not being pregnant, and pregnancy is the indexed event.

The statistical comparison are as follows.

The 24 month survival (not pregnant) rates are 0.56 (56%) for Treatment 1, and 0.19 (19%) for Treatment 2. Putting it the other way round, the pregnancy rate is 44% for Treatment 1 and 81% for Treatment 2

Statistical significance is by the Chi Square Test. Chi Sq = 5.23, df = 1, α p = 0.02, statistically significant

Relative Risk (of pregnancy) Treatment 1 /Treatment 2 = 0.39, 95% Confidence interval = 0.18 to 0.85

In other words, the chance of pregnancy using Treatment 1 is 18% to 85% that of Treatment 2, at the end of 2 years.

Survival analysis

Kaplan EL, Meier P (1958) American Statistical Association Journal. June p. 457-481.

Peto R, Pike MC, Armitage P, et.al., Design and analysis of randomised clinical trials requiring prolonged observations of each patient II. Analysis and examples. Br. J. Cancer 1977 35:1-39.

A good description : Swinscow TDV (revised by MJ Campbell) Statistics at square one 1996 (9th Ed.). BMJ Publishing Group. ISBN 0-7279-0916-9 Chapter 12 Survival analysis. p 114-122.

Sample size

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 176-177