Simple growth models
In the preceding chapters we have defined dynamic models as iterative models, where each next step is based on the preceding step. In this chapter we will introduce and discuss growth models, a type of iterative, dynamic model where the next level of a variable is a function of its preceding value and a growth rate. We will also see that growth processes depend on specific and limited resources. The models we will introduce are classical, general growth models that have been used for explaining phenomena ranging from the growth of insect populations to the replacement of steam by diesel engines. We will explain how growth models can be applied to developmental and learning processes. We will help the reader build his or her own growth models of developmental phenomena.
The first classical model: proportional growth
Proportional growth is as based on an iterative (or recursive) model, in which the change is a proportion of the current state. A simple example is the growth of someone’s capital on a bank account. The bank gives an interest, e.g. 5%, and that is the proportion with which the person’s capital increases. An initial capital of $ 100,- increases with $ 5 the first year, which gives a new capital of $ 105,-, which increases with $ 105*0.05 = $ 5.25 the second year, and so forth. Note the difference with the random walk model that we discussed in the preceding chapter. The random walk model involved an iterative change, just like the current growth model, but the change consisted of a simple (random) addition to the preceding state of the model. In the random walk, the amount of change did not depend on the preceding level (for instance, the number added to the level does not get bigger as the level gets bigger). In the proportional model, the amount of change depends on the level already attained: 5% of a capital of $ 100,- gives an interest of $ 5, but the same 5% gives an interest of $ 50 on a capital of $ 1000,-
In general, we are not used to think of psychological variables or developmental levels in terms of a specific kind of increase, e.g. either simply additional or proportional. For instance, if we think about a child’s increasing social skill, we should ask ourselves if the increase is likely to be additional or proportional. If we assume that it is easier for a child to learn new things about the social world if the child already has a considerable knowledge of that social world, we are implicitly assuming a proportional form of growth for social skills. If we assume that the child increases his or her social skill with some (on average) constant factor, we are assuming an additive model. It is remarkable that, in general, developmental psychologists have so little knowledge of the nature of change in children, in spite of the fact that such change is the core subject of their discipline. The reason for this ignorance is probably that most instances of development are investigated by comparing groups of children of different ages, or by associating the variable under scrutiny with some other variable (for instance family characteristics such as child rearing style). By doing so, we miss a very important aspect of change, namely the way change depends on the level already attained (which amounts to the iterative or recursive nature of change and development that we discussed in the introductory chapter).
The equation for Proportional growth
The equation for proportional growth is very simple
L/t = r.Lequation 2.1
As the reader will recall from the preceding chapter, this equation can be read as follows. The increase in a level, L, over some amount of time, t, equals the level already attained, L, times a ratio or proportion, r. In order to calculate the level at some later time, for instance at time t + t, we simply add the proportional increase to the level already attained
Lt+t = Lt + r.Ltequation 2.2
In the following exercise, we will model a simple proportional growth process and study its properties. We shall assume that our model describes the growth of a child’s lexicon, i.e. the increase in the number of words a child uses and/or understands.
The growth of the lexicon is a well-studied field in developmental psycholinguistics. Although a discussion of lexical growth far extends the scope of the present discussion, we will review some general points. To begin with, the (relatively well-educated) adult’s lexicon is conservatively estimated as comprising about 50,000 words (see for instance Aitchsion, 1994). Let us assume that an 18-year-old person has 50000 words. He or she learned al these words in 18 times 365 days, which is 6570 days. 50,000 divided by 6570 is approximately 7.6. Hence, our 18-year-old has learned between 7 and 8 words a day. It is rather unlikely, however, that lexical growth can be modeled by a simple process of constant addition. We know, for instance, that many young children show a significant spurt in their lexical growth, around the age of 15 to 16 months (see for instance Gillis, 1984, Dromi, 1987). A comparable increase in the rate of word learning is demonstrated by measurements with the McArthur Communicative Development Inventories (Fenson et al., 1993; Fenson et al., 2000). In short, lexical growth is far from a simple linear increase. Let us take a closer look at the data from Esther Dromi’s daughter Keren. Open the file Keren’s lexical data.xls (see also figure Keren’s lexical growth). The data suggest that, the more words Keren knows, the more new ones she learns. It pretty much looks like the growth of a capital on a bank account.
Insert figure Keren’s lexical growth about here.
Building a model of proportional growth
2.1 Open a new Excel file and save it under the name Simple growth models.xls. Rename the first worksheet: replace the name Sheet1 by Proportional growth
see Exceltip rename Worksheet
2.2 Introduce the variables Ini_prop and Rate_prop by means of the Insert Name procedure. Put numerical values in the cells that contain the Ini_prop and rate_prop values (e.g. 1 and 0.1 respectively)
see Exceltip Introduce Named Variables
2.3. The model will be built in the following steps.
2.3.1 In cell D1, write MODEL
2.3.2 In cell D2, write =Ini_prop (see Exceltip Write Equation by selection)
2.3.3. In cell E1, INCREASE
2.3.4. In cell E2 write the equation that specifies the increase part, =D2*Rate_prop.
2.3.5. In cell D3 you will add the preceding state of the model (which is in cell D2) and the increase, which is in cell E2. Thus, in cell D3 write =D2+E2. Copy D3 to D4:D101. First copy E2 to E3 and then copy E3 to E4:E101 (see chapter Random Walk, Step 4; see Exceltip Quick Copy Procedure)
2.3.6. By pasting the model equation to 100 consecutive cells, you have defined a model that contains 100 steps. The meaning of the step can be defined as you wish. For instance, if you interpret each step as a day, your model specifies lexical growth over a period of 100 days.
2.4 Make a line graph of your model, which is comprised in the range D1:D100 (the procedure for making a line graph is described in the Exceltip Make Graphs). If the growth rate is high, your model will produce VERY high levels towards the end, i.e. very high numbers of words in the (imaginary) lexicon. Excel will specify those numbers in scientific notation, for instance 3E+59. This represents a number consisting of a 3 followed by 59 zeroes (which is probably more than the number of atoms in your body…) Experiment with different values for Ini_prop and Rate_prop, but change only one parameter at a time. Press F9 to recalculate your model with the new parameter values.
Assignment: Describe the form of the growth curve. Take a look at the final state of the curve (last point of the model range). Is this a realistic final state, taking into account that the model is supposed to describe lexical growth? What happens if the growth goes on for a considerably longer time?
2.5. The preceding model was a purely deterministic model. In reality, however, the parameters represent variables, mechanisms and factors that fluctuate over time in a random way. For instance, the growth rate of the lexicon will vary from day to day, depending on factors such as motivation, the time the parents spend with the child, and so forth. It is even possible that at some times the growth rate is negative, i.e. that the child forgets more words than it learns. What will be the effect of such random fluctuation of the parameters? We will assume that the parameters – in fact, there is only one parameter that really matters, namely Rate_prop – fluctuate around an average value, with a characteristic standard deviation, and that the fluctuation is symmetric around the average. To put it differently, we will assume that the fluctuation has a normal distribution, with a specified mean and standard deviation.
We will assume that the growth rate, Rate_prop, is the mean of the fluctuating set of rates. Each step (day, week, …) we randomly pick a growth rate from the probabilistic distribution of growth rates.
2.5.1 Introduce a third variable, namely Rate_prop_sd, which will specify the standard deviation of the fluctuating growth rate. In cell A3, write Rate_prop_sd and insert a name according to the procedure specified earlier, see Exceltip Introduce Named Variables). Write a numerical value for this standard deviation in cell B3 (for instance 1/3d of the value of the growth rate itself) (see Exceltip Automatically Expand Columns)
2.5.2. Use column F to assign an arbitrary value to the growth rate for each time step. In cell F1, write VARIABLE GROWTH RATE. In cell F2 we will enter the equation that picks a variable growth rate from a distribution with the predefined mean and standard deviation. We will use a function from Poptools. In cell F2, write =dNormaldev(Rate_prop, rate_prop_sd) (the variable names can be typed directly, or entered by clicking on the cells that contain their values, B2 and B3). If you wish to use an equivalent Excel formula, write =Norminv(Rand(),Rate_prop, Rate_prop_sd). Copy to F3:F101. Note that your model still refers to the constant Rate_prop: you have to change the equations so that they refer to the variable growth rate, which appears in the F-column. To do that, select cell E2, then put the cursor in the formula bar (under the menu). The formula bar works like a small word processor: delete the word Rate_prop, type F2 instead (or select cell F2), then Enter. Copy the altered formula, then paste to E3:E100. Each time you press the F9 button, Excel will recalculate your model with values for rate_prop that are randomly changed at each step of your model (that is, within the specifications of the mean and the standard deviation). You will see that the curve changes each time you press F9. Use a relatively small growth ratio (for instance 0.05) and a standard deviation that is about twice to three times as small as the ratio; compare with standard deviations that are considerably bigger).
2.5.3. We shall now investigate the effect of the randomly varying growth ratio on the end state of our model. The end state can be found in the last step of the model, which should be cell D101. Both Poptools and Paul’s Functions contain a menu option that allows you to automatically recalculate a pre-specified variable (for instance the last cell of your model) as many times as you wish, that keeps the values of each run and gives you a summary statistics of all the values obtained. We shall investigate the values of the randomly varying growth rates and of the resulting end states of the model. In cell G1, write =D101 (cell G1 now refers to the end state of the model). In cell G2, write =Average(F2:F101). This equation calculates the average growth rate over the 100 steps of the model. Press F9 a few times and watch the numbrs change. Activate the menu option Functies/Monte Carlo Simulatie (see chapter 1, Step 10). The first field of the Monte Carlo window should contain a reference to the cells that you want to recalculate with each Monte Carlo run. These are the cells G1:G2. The second field (which eventually compares the Monte Carlo run with empirically observed values) does not need to be specified. The third field contains a reference to an output cell. Click, for instance, on cell H1. Check the option “Keep output on a separate Worksheet”. You will be asked to specify a name for a new worksheet that will contain the 1000 end states and 1000 average growth rates that your Monte Carlo run will produce. Type OP1.
Average / 5020Median / 4593
Minimum / 957
Maximum / 17644
Perc 0.025 / 1813
Perc 0.975 / 10797
StDev / 2335.761
Skewness / 1.277
p-value / 1.000
# of simulations / 1000
2.5.4. After finishing the Monte Carlo run, first take a look at the summary statistics of the run (which will begin in cell H1). In order to make sense of the numbers, recall that the model is supposed to mimic the growth of the child’s lexicon. The table shows a list of possible results fro the first variable, namely the end state of the model. The average number of words learned at the end is 5020. Look at the values specified by Perc 0.025 and Perc. 0.975. Those values form the boundaries of the 95% interval (0.975 minus 0.0025 equals 0.95). That is, 95% of the values obtained in the simulations lie between 1813 words and 10,797 words. Look at the skewness of the distribution, which is 1.277. This means that the set of outcomes is strongly skewed to the right: there’s a long tail on increasing numbers to the right of the distribution’s center. In order to obtain a graphic representation of the distribution, select worksheet OP1, then activate Functies/Maak frequentietabel and type 15 in the text box that asks you for the number of categories (see Chapter 1, Step 5). Describe the distribution of the end states. Compare this with the distribution of the growth rates. Calculate the correlation between the end state and the growth rate. What do you expect to find? Enter the following equation in an empty cell =correl(a2:a1001,b2:b2002), which will calculate the correlation between all growth rates and the corresponding end states. Does the outcome confirm your expectations?
2.6. Let us conclude this section on proportional growth by trying to build a model of lexical growth between the ages of 1 year and 18 years.
Assignment: Assume that the person learns his first word at the age of 1 year, and knows 50,000 words at the age of 18 years. Model lexical growth in steps of one month (every cell of your model will represent a month in the life of our subject). What is the monthly growth rate, provided the subject must know about 50,000 words at the age of 18? Assume that the subject’s language contains 200,000 words in total (an imaginary number). With what growth rate can the subject have learned all the words in his language by the age of 18? And how many words (if they were available) would he have learned with a growth rate that were 10% bigger?
Solution of Assignment 2.6
A short historical note
The model we have just worked with has already a long history. It was introduced in 1798 by Thomas Robert Malthus (1766-1834), an English economist. According to Malthus, the model of exponential growth described the growth of the human population. Malthus was well aware of the fact that all growth must be supported by resources, for instance, food and housing in the case of humans. Malthus predicted that such commodities would grow in an additive way. Such growth cannot keep up with the exponential growth of the human population. Malthus foresaw an overpopulated earth, with people fighting over scarce food supplies. He pleaded for an active policy of birth control in order to avoid such disaster. The growth rate r in the proportional growth model is still called the Malthusean term
A second classical growth model: restricted growth
A striking property of the preceding model was the absence of any limit on the growth of the variable at issue. Relatively small increases in the growth rate could cause the process to reach unrealistically high levels.
In the real world, growth depends on resources. However abundant they may be, they are always limited. For instance, a language contains many words, but a person cannot learn more words than there are words in his or her language. In practice, however, we learn considerably less words than the number of words that our mother tongue possesses. Many words will occur so rarely that we will probably never be confronted with them. Other words belong to highly specialized regions of activity or profession, which we will most likely never participate in. Some people are simply more verbally talented than others (whatever that talent may mean in practice) and will therefore learn words more easily. Others take a high interest in language, and actively seek for new words (for instance, people who like to do crossword puzzles). All these properties, some of which are in the environment and some of which are in the person himself, form the resources for word learning and will determine how many words, approximately, a person will actually learn during the person’s lifetime.
In the first model, that of proportional growth, the notion of resource was inexistent. The second model, however, considers the limited resource as the only factor that determines the growth process (in addition to the growth rate, that is, but the growth rate as such is not considered a resource factor).